{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":51322,"title":"Solve an ODE: diffusion problem 1","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 264.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 132.125px; transform-origin: 407px 132.125px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 295.017px 7.91667px; transform-origin: 295.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f''+a eta f' = 0\" style=\"width: 91px; height: 18px;\" width=\"91\" height=\"18\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 298.233px 7.91667px; transform-origin: 298.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.1167px 7.91667px; transform-origin: 45.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The problem has the conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(0) = f0\" style=\"width: 60px; height: 20px;\" width=\"60\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIIAAAAmCAYAAAAIjkMFAAADx0lEQVR4nO2babGrQBCFjwccxAAGogAFOIiDOIiFaEACHmIBDVjI/TFzioYHTPew581XlbobMNuZ3pgLJBKJRCKRSCQSO3ADcDfekwF4+69HUQB4HNj+z5ADaADUcIvaQjexmb8n365rap5wff8Vcrg1ePqvt60bvAP4Aqj8zw//c4v5XZ4B+MBuQbbkDeB1dCcWksNtrhbO0t3hxPCFG9smljfzDbboFFf4RuvAvRXOOekN3BiuSA63Fl/8awEohhobiIEPr0IXDigRthhHccd5+xbig27nj9H4v68eD1F9pfG+Bk5EZ+WD68ULJdxafDHtbl/o3PZqMYNs2PJQ3nfmHXeFPg6hNZizZnTbMZt3kso/sDHe90E4fjiaDBuZ0I24oVvgj/I6qzvvwXTkic4tNOJ3D8zvInYkxi0Uop0i0A7QRctPOPVbd/cVBEuYuWkWmNe1SxrkxL4HDcsJn4Mm15IyFuhEJz9T0T1rGsPr24nrp+AYYyjRzcmSjxYG7RYhWF16sGHLovI+bQGJwql9O6X/Xg5GLq5Mn6Y+WjFY+yoZ9jHmY9mxcj1CQa6cn8WFPMYH1jSLE6Th5p8/pnA5cC5uhs4SvNAN8ja4vlH2mYWxmILXGhbBEp/I8YXukyJdLASqyupDLULg4KbMF1MhuRPmdrzMcjRWgX73CgGjFEKoSCfd5iIhyMjTWhm0CIFlUs3zrLtB438phDPXOwit164xgsxFraVYixA0fpLuQxuvcMI0qdOVhLB71gD0TbI1JbMKQaNauRs+gT5xwjRVw5gMR967Z9bAukfIXWuvU8HFnCtcTEERaQREXxaakGHgOOeuLH6fz40Rwt5ZA9CvLE4hLcdiS6dNU8awROIyM5kKarhreW3IZbF9jRAp2hj2zhrYZshFLrHmPXLxoJhaNe/XDFIOrEG/OligX+LO0HcRY2LgK3OtgGvEWb0j0b59XGwN5GTHRpwN9IvBgc2ZT9mP9+DvLzhBlL7dUAxB6EvPeF5iDhbUxqwo126V8wicaOuLJskLev/Ho2xjIqgxLsYHxquL1cT1YzAz2vx41wawxE4rKk8oVVjpjWrI9GhgHcLiWuT5uxLhQkiGzk9rrh9S4TovnKYYvnRbTdQy9VhamnzhvBNNoZ7hQO0pYfC2RgDF9wKrHY5YkRrXiw02g/9rIKN7RulrnTqmLzuTH37gepnCpsgiTYYu7Vv7HF+BjU7VRpBDn1X8NzANofm2pHwxbR0thhwrRtS/xg3OVO7xXzJHL8CZ3FMikUgkEolEInE+/gCQzs5E34QVuwAAAABJRU5ErkJggg==\" alt=\"f(infinity) = 0\" style=\"width: 65px; height: 19px;\" width=\"65\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 372.133px 7.91667px; transform-origin: 372.133px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = diffusion1ODE(eta,a,f0)\r\n%  eta = independent variable\r\n%  a   = constant\r\n%  f0  = value of f at eta = 0\r\n\r\n   f = g(eta,a,f0)\r\nend","test_suite":"%%\r\na = 1/2; \r\nf0 = 1;\r\neta = 0:0.2:1;\r\nf_correct = [1 0.887537083981715 0.777297410789522 0.671373240540873 0.571607644953331 0.479500122186953];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2; \r\nf0 = 1;\r\neta = 0.5;\r\nf_correct = 0.479500122186953;\r\nassert(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 1;\r\nf0 = 0.5;\r\neta = [0.12 0.23 0.456 0.789 1.011];\r\nf_correct = [0.452241573979416 0.409045884857994 0.324194989107724 0.215056003266342 0.156008214896009];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2;\r\nf0 = 1;\r\neta = 1:4;\r\nf_correct = [0.157299207050285 0.004677734981047 2.209049699858544e-05 1.541725790028002e-08];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 3/4;\r\nf0 = 4/3;\r\neta = 3*logspace(-2,0,3);\r\nf_correct = [1.305696910508165 1.060016229285651 0.012499691279247];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 0.01;\r\nf0 = 1;\r\neta = rand/120;\r\nf_correct = polyval(flip([1 -0.0797885 0 0.000132981 0 -1.99471e-7]),eta);\r\nf = diffusion1ODE(eta,a,f0);\r\nassert(all(abs(f-f_correct)\u003c1e-7))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-04-08T01:02:26.000Z","updated_at":"2025-05-04T20:55:44.000Z","published_at":"2021-04-08T01:09:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f''+a eta f' = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime + a\\\\eta f\\\\prime = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The problem has the conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(0) = f0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(0) = f_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(infinity) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51625,"title":"Solve an ODE: diffusion problem 2","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 317px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 158.5px; transform-origin: 407px 158.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 7.91667px; transform-origin: 63.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 295.017px 7.91667px; transform-origin: 295.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f''+a(f+eta f') = 0\" style=\"width: 130px; height: 19px;\" width=\"130\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 298.233px 7.91667px; transform-origin: 298.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44.3333px 7.91667px; transform-origin: 44.3333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and its derivatives vanish at infinity, and it is subject to the constraint\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"integral(f,{eta,-inf,inf}) = 1\" style=\"width: 78px; height: 44px;\" width=\"78\" height=\"44\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 349.967px 7.91667px; transform-origin: 349.967px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves diffusion of a quantity instantaneously injected at a point—for example, a spill of a contaminant in an initially clean river. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = diffusion2ODE(eta,a)\r\n%  eta = independent variable\r\n%  a   = positive constant\r\n  f = a*sqrt(eta);\r\nend","test_suite":"%%\r\na = 1/2; \r\neta = 0:0.2:1;\r\nf_correct = [0.282094791773878 0.279287901697234 0.271033696776216 0.257815227404741 0.240385324709827 0.219695644733861];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 2; \r\neta = 0.5;\r\nf_correct = 0.439391289467722;\r\nassert(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13)\r\n\r\n%%\r\na = 1;\r\neta = [0.12 0.23 0.456 0.789 1.011];\r\nf_correct = [0.396080211793656 0.388528585315836 0.359548380672770 0.292234407541508 0.239309153606614];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 2;\r\neta = 1:4;\r\nf_correct = [0.207553748710297 0.010333492677046 0.000069626525973 0.000000063491173];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 3/4;\r\neta = 3*logspace(-2,0,3);\r\nf_correct = [0.345377564870647 0.334028296534584 0.011822159682599];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 1;\r\neta = rand/120;\r\nf_correct = polyval([1/10321920 0 -1/645120 1/46080 0 -1/3840 0 1/384 0 -1/48 0 1/8 0 -1/2 0 1]./(sqrt(2*pi)),eta);\r\nf = diffusion2ODE(eta,a);\r\nassert(all(abs(f-f_correct)\u003c1e-13))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-01T14:31:00.000Z","updated_at":"2021-05-01T14:34:52.000Z","published_at":"2021-05-01T14:34:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f''+a(f+eta f') = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime+a(f+\\\\eta f\\\\prime)=0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and its derivatives vanish at infinity, and it is subject to the constraint\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"integral(f,{eta,-inf,inf}) = 1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\int_{-\\\\infty}^\\\\infty fd\\\\eta=1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves diffusion of a quantity instantaneously injected at a point—for example, a spill of a contaminant in an initially clean river. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51087,"title":"Solve an ODE: equation for a 2D laminar jet","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 503px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 251.5px; transform-origin: 407px 251.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 330.55px 7.91667px; transform-origin: 330.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution for a two-dimensional laminar jet, the following nonlinear ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f’’’ + 2(f’2+ff”) = 0\" style=\"width: 147.5px; height: 20px;\" width=\"147.5\" height=\"20\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 236.783px 7.91667px; transform-origin: 236.783px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 103.467px 7.91667px; transform-origin: 103.467px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The problem has the conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(0) = f''(0) = 0\" style=\"width: 115px; height: 19px;\" width=\"115\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f'(oo) = 0\" style=\"width: 69px; height: 19px;\" width=\"69\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75.0417px 7.91667px; transform-origin: 75.0417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the constraint         \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"integral of f'^2 from -infinity to infinity = a\" style=\"width: 119.5px; height: 44px;\" width=\"119.5\" height=\"44\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 71.8417px 7.91667px; transform-origin: 71.8417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                                     \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 43.1667px 7.91667px; transform-origin: 43.1667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a constant.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 103.575px 7.91667px; transform-origin: 103.575px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The test suite allows MATLAB’s functions for numerical solution of ODEs, but the equation has a relatively simple analytical solution. If you would like a hint for the analytical solution, execute this command:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 296.25px 7.91667px; transform-origin: 296.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003echar('Zxj%ymj%uwtizhy%wzqj%tk%inkkjwjsynfynts%y|t%ynrjx%fsi%nsyjlwfyj%ymwjj%ynrjx3'-5)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 147px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 73.5px; text-align: left; transform-origin: 384px 73.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 129.925px 7.91667px; transform-origin: 129.925px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves a jet in the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"x, y\" style=\"width: 24.5px; height: 18px;\" width=\"24.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 108.133px 7.91667px; transform-origin: 108.133px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e plane emanating from a source at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"(x,y) = (0,0)\" style=\"width: 87.5px; height: 19px;\" width=\"87.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.1667px 7.91667px; transform-origin: 64.1667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. It can be solved by expressing the velocities \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 278.117px 7.91667px; transform-origin: 278.117px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in terms of a streamfunction and employing a similarity solution that combines the spatial coordinates into a single variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73.5083px 7.91667px; transform-origin: 73.5083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. In the problem above, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(eta)\" style=\"width: 32px; height: 19px;\" width=\"32\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 172.308px 7.91667px; transform-origin: 172.308px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is proportional to the streamfunction. The conditions at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"eta = 0\" style=\"width: 36px; height: 18px;\" width=\"36\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.4583px 7.91667px; transform-origin: 47.4583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e define the line \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y = 0\" style=\"width: 37px; height: 18px;\" width=\"37\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143.508px 7.91667px; transform-origin: 143.508px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e as a line of symmetry; the transverse velocity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.733px 7.91667px; transform-origin: 128.733px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the shear stress are zero there. The condition \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f'(oo) = 0\" style=\"width: 69px; height: 19px;\" width=\"69\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.45px 7.91667px; transform-origin: 110.45px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e states that the streamwise velocity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 203.767px 7.91667px; transform-origin: 203.767px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is zero far from the source. The integral constraint states that the momentum flux of the jet is constant. The value \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"a = 4/3\" style=\"width: 51.5px; height: 19px;\" width=\"51.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 200.7px 7.91667px; transform-origin: 200.7px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e corresponds to the physical problem of the jet. Other values are included for variety.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = lamjetODE(eta,a)\r\n  f = g(eta,a);\r\nend","test_suite":"%%\r\na = 4/3; \r\neta = 0:0.2:1;\r\nf_correct = [0 0.197375320224904 0.379948962255225 0.537049566998035 0.664036770267849 0.761594155955765];\r\nassert(all(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 4/3; \r\neta = log(2);\r\nf_correct = 0.6;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 1;\r\neta = pi;\r\nf_correct = 0.902552583791843;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 0.5;\r\neta = 1.5;\r\nf_correct = 0.572446107496431;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 0.75;\r\neta = -4:0;\r\nf_correct = [-0.823247562011528 -0.813902901498664 -0.766864130068021 -0.559711742572462 0];\r\nassert(all(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = rand;\r\neta = 4*rand;\r\nassert(abs(lamjetODE(eta,a)+lamjetODE(-eta,a))\u003c1e-10)","published":true,"deleted":false,"likes_count":2,"comments_count":2,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2021-03-21T03:15:00.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-03-21T02:07:31.000Z","updated_at":"2021-03-22T00:42:37.000Z","published_at":"2021-03-21T02:24:49.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution for a two-dimensional laminar jet, the following nonlinear ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f’’’ + 2(f’2+ff”) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime\\\\prime + 2(f\\\\prime^2+ff\\\\prime\\\\prime) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The problem has the conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(0) = f''(0) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(0) = f\\\\prime\\\\prime(0) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f'(oo) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the constraint         \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"integral of f'^2 from -infinity to infinity = a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\int_{-\\\\infty}^\\\\infty\\\\left[f\\\\prime(\\\\eta)\\\\right]^2d\\\\eta=a\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e                                     \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a constant.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The test suite allows MATLAB’s functions for numerical solution of ODEs, but the equation has a relatively simple analytical solution. If you would like a hint for the analytical solution, execute this command:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003echar('Zxj%ymj%uwtizhy%wzqj%tk%inkkjwjsynfynts%y|t%ynrjx%fsi%nsyjlwfyj%ymwjj%ynrjx3'-5)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves a jet in the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x, y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex, y\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e plane emanating from a source at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(x,y) = (0,0)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(x,y) = (0,0)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. It can be solved by expressing the velocities \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in terms of a streamfunction and employing a similarity solution that combines the spatial coordinates into a single variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. In the problem above, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(eta)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(\\\\eta)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is proportional to the streamfunction. The conditions at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e define the line \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as a line of symmetry; the transverse velocity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the shear stress are zero there. The condition \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f'(oo) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e states that the streamwise velocity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is zero far from the source. The integral constraint states that the momentum flux of the jet is constant. The value \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a = 4/3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea = 4/3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e corresponds to the physical problem of the jet. Other values are included for variety.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":51322,"title":"Solve an ODE: diffusion problem 1","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 264.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 132.125px; transform-origin: 407px 132.125px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 295.017px 7.91667px; transform-origin: 295.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f''+a eta f' = 0\" style=\"width: 91px; height: 18px;\" width=\"91\" height=\"18\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 298.233px 7.91667px; transform-origin: 298.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.1167px 7.91667px; transform-origin: 45.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The problem has the conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(0) = f0\" style=\"width: 60px; height: 20px;\" width=\"60\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(infinity) = 0\" style=\"width: 65px; height: 19px;\" width=\"65\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 372.133px 7.91667px; transform-origin: 372.133px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = diffusion1ODE(eta,a,f0)\r\n%  eta = independent variable\r\n%  a   = constant\r\n%  f0  = value of f at eta = 0\r\n\r\n   f = g(eta,a,f0)\r\nend","test_suite":"%%\r\na = 1/2; \r\nf0 = 1;\r\neta = 0:0.2:1;\r\nf_correct = [1 0.887537083981715 0.777297410789522 0.671373240540873 0.571607644953331 0.479500122186953];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2; \r\nf0 = 1;\r\neta = 0.5;\r\nf_correct = 0.479500122186953;\r\nassert(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 1;\r\nf0 = 0.5;\r\neta = [0.12 0.23 0.456 0.789 1.011];\r\nf_correct = [0.452241573979416 0.409045884857994 0.324194989107724 0.215056003266342 0.156008214896009];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2;\r\nf0 = 1;\r\neta = 1:4;\r\nf_correct = [0.157299207050285 0.004677734981047 2.209049699858544e-05 1.541725790028002e-08];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 3/4;\r\nf0 = 4/3;\r\neta = 3*logspace(-2,0,3);\r\nf_correct = [1.305696910508165 1.060016229285651 0.012499691279247];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 0.01;\r\nf0 = 1;\r\neta = rand/120;\r\nf_correct = polyval(flip([1 -0.0797885 0 0.000132981 0 -1.99471e-7]),eta);\r\nf = diffusion1ODE(eta,a,f0);\r\nassert(all(abs(f-f_correct)\u003c1e-7))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-04-08T01:02:26.000Z","updated_at":"2025-05-04T20:55:44.000Z","published_at":"2021-04-08T01:09:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f''+a eta f' = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime + a\\\\eta f\\\\prime = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The problem has the conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(0) = f0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(0) = f_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(infinity) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51625,"title":"Solve an ODE: diffusion problem 2","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 317px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 158.5px; transform-origin: 407px 158.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 7.91667px; transform-origin: 63.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 295.017px 7.91667px; transform-origin: 295.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f''+a(f+eta f') = 0\" style=\"width: 130px; height: 19px;\" width=\"130\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 298.233px 7.91667px; transform-origin: 298.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44.3333px 7.91667px; transform-origin: 44.3333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and its derivatives vanish at infinity, and it is subject to the constraint\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"integral(f,{eta,-inf,inf}) = 1\" style=\"width: 78px; height: 44px;\" width=\"78\" height=\"44\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 349.967px 7.91667px; transform-origin: 349.967px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves diffusion of a quantity instantaneously injected at a point—for example, a spill of a contaminant in an initially clean river. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = diffusion2ODE(eta,a)\r\n%  eta = independent variable\r\n%  a   = positive constant\r\n  f = a*sqrt(eta);\r\nend","test_suite":"%%\r\na = 1/2; \r\neta = 0:0.2:1;\r\nf_correct = [0.282094791773878 0.279287901697234 0.271033696776216 0.257815227404741 0.240385324709827 0.219695644733861];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 2; \r\neta = 0.5;\r\nf_correct = 0.439391289467722;\r\nassert(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13)\r\n\r\n%%\r\na = 1;\r\neta = [0.12 0.23 0.456 0.789 1.011];\r\nf_correct = [0.396080211793656 0.388528585315836 0.359548380672770 0.292234407541508 0.239309153606614];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 2;\r\neta = 1:4;\r\nf_correct = [0.207553748710297 0.010333492677046 0.000069626525973 0.000000063491173];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 3/4;\r\neta = 3*logspace(-2,0,3);\r\nf_correct = [0.345377564870647 0.334028296534584 0.011822159682599];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 1;\r\neta = rand/120;\r\nf_correct = polyval([1/10321920 0 -1/645120 1/46080 0 -1/3840 0 1/384 0 -1/48 0 1/8 0 -1/2 0 1]./(sqrt(2*pi)),eta);\r\nf = diffusion2ODE(eta,a);\r\nassert(all(abs(f-f_correct)\u003c1e-13))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-01T14:31:00.000Z","updated_at":"2021-05-01T14:34:52.000Z","published_at":"2021-05-01T14:34:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f''+a(f+eta f') = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime+a(f+\\\\eta f\\\\prime)=0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and its derivatives vanish at infinity, and it is subject to the constraint\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"integral(f,{eta,-inf,inf}) = 1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\int_{-\\\\infty}^\\\\infty fd\\\\eta=1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves diffusion of a quantity instantaneously injected at a point—for example, a spill of a contaminant in an initially clean river. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51087,"title":"Solve an ODE: equation for a 2D laminar jet","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 503px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 251.5px; transform-origin: 407px 251.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 330.55px 7.91667px; transform-origin: 330.55px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution for a two-dimensional laminar jet, the following nonlinear ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f’’’ + 2(f’2+ff”) = 0\" style=\"width: 147.5px; height: 20px;\" width=\"147.5\" height=\"20\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 236.783px 7.91667px; transform-origin: 236.783px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 103.467px 7.91667px; transform-origin: 103.467px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The problem has the conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(0) = f''(0) = 0\" style=\"width: 115px; height: 19px;\" width=\"115\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f'(oo) = 0\" style=\"width: 69px; height: 19px;\" width=\"69\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75.0417px 7.91667px; transform-origin: 75.0417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the constraint         \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"integral of f'^2 from -infinity to infinity = a\" style=\"width: 119.5px; height: 44px;\" width=\"119.5\" height=\"44\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 71.8417px 7.91667px; transform-origin: 71.8417px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                                     \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 43.1667px 7.91667px; transform-origin: 43.1667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a constant.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 103.575px 7.91667px; transform-origin: 103.575px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The test suite allows MATLAB’s functions for numerical solution of ODEs, but the equation has a relatively simple analytical solution. If you would like a hint for the analytical solution, execute this command:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 296.25px 7.91667px; transform-origin: 296.25px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003echar('Zxj%ymj%uwtizhy%wzqj%tk%inkkjwjsynfynts%y|t%ynrjx%fsi%nsyjlwfyj%ymwjj%ynrjx3'-5)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 147px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 73.5px; text-align: left; transform-origin: 384px 73.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 129.925px 7.91667px; transform-origin: 129.925px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves a jet in the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"x, y\" style=\"width: 24.5px; height: 18px;\" width=\"24.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 108.133px 7.91667px; transform-origin: 108.133px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e plane emanating from a source at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"(x,y) = (0,0)\" style=\"width: 87.5px; height: 19px;\" width=\"87.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.1667px 7.91667px; transform-origin: 64.1667px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. It can be solved by expressing the velocities \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 278.117px 7.91667px; transform-origin: 278.117px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in terms of a streamfunction and employing a similarity solution that combines the spatial coordinates into a single variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 73.5083px 7.91667px; transform-origin: 73.5083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. In the problem above, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(eta)\" style=\"width: 32px; height: 19px;\" width=\"32\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 172.308px 7.91667px; transform-origin: 172.308px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is proportional to the streamfunction. The conditions at \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"eta = 0\" style=\"width: 36px; height: 18px;\" width=\"36\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.4583px 7.91667px; transform-origin: 47.4583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e define the line \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"y = 0\" style=\"width: 37px; height: 18px;\" width=\"37\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 143.508px 7.91667px; transform-origin: 143.508px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e as a line of symmetry; the transverse velocity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ev\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.733px 7.91667px; transform-origin: 128.733px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and the shear stress are zero there. The condition \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIoAAAAmCAYAAAAbWQPxAAAD3klEQVR4nO2aa7GrMBSFlwccYKAGquAowAEOcFAL1VAJeMACGmqB+wPWsMvNYwcSCDP5Zpg5PU0hj521HwQoFAqFQqFQKBQKqAE8le3qgPs+ALx29SgeHXRjKzh4ABgB9ADeAL4AWkfbCcAQcO8eQHWwj0epAHwANBf3IxYVgD/MG6Bb/k7KE/PCf5bP7fL5C/Pivpbv34p70wCvNhJSYTbw5JOamA7z+rwxr98f5nElG1u1PPCL1ZX8YTaE3vKbcfn+obj3iPwWpcbveO/GG+aNyrWckGDOO/yqiQ+6HU37N+zGdjUd8u2bC27iCWaVbpbvoqs4LVDrt+mWfGpSL+1yDR4rhI07F6jmNiPnuCZETB5ofRP0MjxArybaYPcq7tBHiVSTztFuwKoqUfgE3pDW6pM0tnMNJgc48T51zAXGJr4YRLbbPbYWazpFtzOK/7WwG0IDnZxRqULdTrX8ln3x/f5oengXgyY9dAbQiXa2EocXTqq0uo/4v89na1wU7x3ar8lw9TBPSoPV0OU1WNrbYO1oD12EKyRGkuPUGoqmhOFE3ix2wNljXkQtNKzX0pcWa9BGxZMTI2Mr07VtH7OvElcftJc229w+z7VhW+y7vxHGJ7ai2hFcUfkWxglbiWQVdbv4TM9ZOeaEPTbttRPE3+whhqKEuEuOzWfYT0Q0FMp2ilpCyH17uINpRvA0FiqAbUdRnbQFNSrrHQJaaSiusUnFPWQorHFEzbUFIYbiG4ysNmr8s2yvcak0lFzrPRI5D6fEKDIfT1Fe1xrK9h2TDdnfEX6loDvRRPx3MpRTsx5gfamnqYnsIdRQNHUcGX/4dgknSpNR7MnQ5HPOzHrkurkMW9vOCy0zVVWSbzF9SBfoG5DcTT4lDIk7eN89nJ31SGV1KQXHdDhRiZZjWwjJJOh3B9gHxV1vyoJsz9duggHX1lFCXf9p73qYXmqleQ/aF4fAb+GvxzpxrNIy6+HO8xkL3Zl2bKkC+lRQVWyZT4NIaiKLManOY9ClaBaLZ0Nc8ixPx/HgkUwVWe6Xh3k0cNJzOy/jw3UehYpzeEx8SLQ3ixa0b5mB9RScyUhe+H9nVPgN2OTVGdrbeCH9PKRCnjKUJ9y+iGT43I2p5ZYSGKJaVAX6bt9va6wvOTXtt1CN7kqNsJeoamSgc0YlckC+C9Egr7O8WcFdftZhHQbOuZ1NZfX2brFJEirM/kvm28wYzqxCMnPJiQ/ulekkRZZxK6y7O1XtxNeXK55rokU+fcmCB2Z5HbH64ysnqLn4+exDURIDzArkuY0ruTpwzGEOCoVCoVAoFAqFwh7+ATX923/I0qEaAAAAAElFTkSuQmCC\" alt=\"f'(oo) = 0\" style=\"width: 69px; height: 19px;\" width=\"69\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.45px 7.91667px; transform-origin: 110.45px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e states that the streamwise velocity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eu\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 203.767px 7.91667px; transform-origin: 203.767px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is zero far from the source. The integral constraint states that the momentum flux of the jet is constant. The value \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"a = 4/3\" style=\"width: 51.5px; height: 19px;\" width=\"51.5\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 200.7px 7.91667px; transform-origin: 200.7px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e corresponds to the physical problem of the jet. Other values are included for variety.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = lamjetODE(eta,a)\r\n  f = g(eta,a);\r\nend","test_suite":"%%\r\na = 4/3; \r\neta = 0:0.2:1;\r\nf_correct = [0 0.197375320224904 0.379948962255225 0.537049566998035 0.664036770267849 0.761594155955765];\r\nassert(all(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 4/3; \r\neta = log(2);\r\nf_correct = 0.6;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 1;\r\neta = pi;\r\nf_correct = 0.902552583791843;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 0.5;\r\neta = 1.5;\r\nf_correct = 0.572446107496431;\r\nassert(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 0.75;\r\neta = -4:0;\r\nf_correct = [-0.823247562011528 -0.813902901498664 -0.766864130068021 -0.559711742572462 0];\r\nassert(all(abs(lamjetODE(eta,a)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = rand;\r\neta = 4*rand;\r\nassert(abs(lamjetODE(eta,a)+lamjetODE(-eta,a))\u003c1e-10)","published":true,"deleted":false,"likes_count":2,"comments_count":2,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2021-03-21T03:15:00.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-03-21T02:07:31.000Z","updated_at":"2021-03-22T00:42:37.000Z","published_at":"2021-03-21T02:24:49.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution for a two-dimensional laminar jet, the following nonlinear ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f’’’ + 2(f’2+ff”) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime\\\\prime + 2(f\\\\prime^2+ff\\\\prime\\\\prime) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The problem has the conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(0) = f''(0) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(0) = f\\\\prime\\\\prime(0) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f'(oo) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the constraint         \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"integral of f'^2 from -infinity to infinity = a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\int_{-\\\\infty}^\\\\infty\\\\left[f\\\\prime(\\\\eta)\\\\right]^2d\\\\eta=a\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e                                     \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a constant.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The test suite allows MATLAB’s functions for numerical solution of ODEs, but the equation has a relatively simple analytical solution. If you would like a hint for the analytical solution, execute this command:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003echar('Zxj%ymj%uwtizhy%wzqj%tk%inkkjwjsynfynts%y|t%ynrjx%fsi%nsyjlwfyj%ymwjj%ynrjx3'-5)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves a jet in the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x, y\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex, y\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e plane emanating from a source at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"(x,y) = (0,0)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(x,y) = (0,0)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. It can be solved by expressing the velocities \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in terms of a streamfunction and employing a similarity solution that combines the spatial coordinates into a single variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. In the problem above, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(eta)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(\\\\eta)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is proportional to the streamfunction. The conditions at \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e define the line \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"y = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as a line of symmetry; the transverse velocity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"v\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the shear stress are zero there. The condition \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f'(oo) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e states that the streamwise velocity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"u\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is zero far from the source. The integral constraint states that the momentum flux of the jet is constant. The value \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a = 4/3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea = 4/3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e corresponds to the physical problem of the jet. Other values are included for variety.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"term":"tag:\"similarity 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