symbolic integrate and plot a function of one variable

Dear Hello

I apologize if there are any inaccuracies in the question.

I'm having a problem integrating a differential equation in a single variable.

The following is the differential equation satisfied by the function "x(u)" that is the x-component of a 2D curve given in parametric form (...this curve is the pathline of an infinitesimal volume of fluid, but that doesn't matter). The parameter "u" is time.

So pathline = {x(u), y(u)} is a 2D curve for 'u' in a certain interval.

Equation:

diff( x(u),u ) == 159648450049389632/(3991211251234741*(4 - (C1 + 2*u + 3)^(1/2))^(1/2)) - 8*x(u) + 14

C1 is a parameter that I'd like to keep,

and I look for a solution in the interval of 'u' in which the above expression is real.

I proceed by doing

dsolve( diff( x(u),u ) == 159648450049389632/(3991211251234741*(4 - (C1 + 2*u + 3)^(1/2))^(1/2)) - 8*x(u) + 14)

I get x(u) like this:

x(u) = exp(-8*u)*int(14*exp(8*u) + (159648450049389632*exp(8*u))/(3991211251234741*(4 - (C1 + 2*u + 3)^(1/2))^(1/2)), u, 'IgnoreSpecialCases', true, 'IgnoreAnalyticConstraints', true) + C2*exp(-8*u)

unless I've made a mistake somewhere.

I already had the 'y(u)' component of the pathline as a function of 'u', and it's:

y(u) = -1/(C1 + 2*u)

Now I'd like to visualize this pathline. To do this, I'll first set the two integration constants C1 and C2, e.g. C1=-3 and C2=1,

C1=-3

C2=1

x=eval(x)

y=eval(y)

now both x and y depend only on 'u', so I use the command

fplot(x,y)

but I get an error... I was wondering how to create this graph

thankYou very much !

Val

[Walter Roberson]

syms x(u) y(u) C1 C2
RHS = 159648450049389632/(3991211251234741*(4 - (C1 + 2*u + 3)^(1/2))^(1/2)) - 8*x(u) + 14;
LHS = diff(x,u);
eqn = LHS == RHS

eqn(u) = 

X(u) = dsolve(eqn)

X(u) = 

y(u) = -1/(C1 + 2*u)

y(u) = 

●

C1 = sym(-3)

C1 = 

C2 = sym(1)

C2 = 

1

X1 = subs(X)

X1(u) = 

Y1 = subs(y)

Y1(u) = 

fplot(X1);

Warning: Error in state of SceneNode.
First argument must be an integral 'int(..., ...)'.

Error using mupadengine/evalin2sym
First argument must be an integral 'int(..., ...)'.

Error in mupadengine/feval2sym_NaNsingularity

Error in matlab.graphics.function.internal.sym2fn (line 32)
f2 = feval2sym_NaNsingularity(symengine,'symobj::unapplyPlotFunction',sf,var_f);
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Error in matlab.graphics.function.FunctionLine>getFunction

Error in matlab.graphics.function.FunctionLine/updateFunction

Error in matlab.graphics.function.FunctionLine

Error in fplot>singleFplot (line 268)
hObj = matlab.graphics.function.FunctionLine(fn{1},extraOpts{:},args{:});
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Error in fplot>@(f)singleFplot(cax,{f},limits,extraOpts,args) (line 223)
hObj = cellfun(@(f) singleFplot(cax,{f},limits,extraOpts,args),fn{1},'UniformOutput',false);
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Error in fplot>vectorizeFplot (line 223)
hObj = cellfun(@(f) singleFplot(cax,{f},limits,extraOpts,args),fn{1},'UniformOutput',false);
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Error in fplot (line 186)
hObj = vectorizeFplot(cax,fn,limits,extraOpts,args);
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

hold on
fplot(Y1)
hold off

Answer 1

[Torsten]

syms x(u) y(u) g(u) C1 C2 v uref
g(u) = 159648450049389632/(3991211251234741*(4 - (C1 + 2*u + 3)^(1/2))^(1/2)) + 14;
x = C2*exp(-8*u)+exp(-8*u)*int(exp(8*v)*g(v),v,uref,u);
xs = subs(x,[C1 C2 uref],[-3 1 0]);   % x satisfies x(uref) = C2*exp(-8*uref), thus here x(0) = C2
y = -1/(C1 + 2*u);
ys = subs(y,C1,-3);
fplot(xs,ys,[0 1])

●

[Val]

thankyou Mr Torsten , transforming the indefinite integral to definite integral ---> fplot work ! very powerful plot command

You chose uref = '0' for some reason ?

Hello

Valerio

[Val]

sorry , i saw just now , " % x satisfies x(uref) = C2*exp(-8*uref), thus here x(0) = C2 " thanks 😊

Answer 2

[Walter Roberson]

The basic problem is that the integral that is produced by dsolve() is an indefinite integral, not a definite integral. That works out in theory, but fplot has trouble with it.

[Val]

Ok , i need to transform it to a definite integral...thankyou very much 😊

[Val]

Dear Mr Roberson

there exist an interpreter that transform expression written in matlab language to

expression with mathematical symbols like You done ?

"

"