dsolve

Solve system of differential equations

Syntax

S = dsolve(eqn)

S = dsolve(eqn,cond)

S = dsolve(___,Name=Value)

[y1,...,yN] = dsolve(___)

Description

S = dsolve(eqn) solves the differential equation eqn, where eqn is a symbolic equation. Use diff and == to represent differential equations. For example, diff(y,x) == y represents the equation dy/dx = y. Solve a system of differential equations by specifying eqn as a vector of those equations.

S = dsolve(eqn,cond) solves eqn with the initial or boundary condition cond.

S = dsolve(___,Name=Value) uses additional options specified by one or more Name=Value arguments.

[y1,...,yN] = dsolve(___) assigns the solutions to the variables y1,...,yN.

Examples

Solve Differential Equation

Solve the first-order differential equation $\frac{\mathit{dy}}{\mathit{dt}}=\mathit{ay}$.

Specify the first-order derivative by using diff and the equation by using ==. Then, solve the equation by using dsolve.

syms y(t) a
eqn = diff(y,t) == a*y;
S = dsolve(eqn)

S = $$C_1\hspace{0.17em}{\mathrm{e}}^{a\hspace{0.17em}t}$$

The solution includes a constant. To eliminate constants, see Solve Differential Equations with Conditions. For a full workflow, see Solve Partial Differential Equation of Tsunami Model.

Solve Second-Order Differential Equation

Solve the second-order differential equation $\frac{{\mathit{d}}^2\mathit{y}}{{\mathit{dt}}^2}=\mathit{ay}$.

Specify the second-order derivative of y by using diff(y,t,2) and the equation by using ==. Then, solve the equation by using dsolve.

syms y(t) a
eqn = diff(y,t,2) == a*y;
ySol(t) = dsolve(eqn)

ySol(t) = $$C_1\hspace{0.17em}{\mathrm{e}}^{-\sqrt{a}\hspace{0.17em}t}+C_2\hspace{0.17em}{\mathrm{e}}^{\sqrt{a}\hspace{0.17em}t}$$

Solve Differential Equations with Conditions

Solve the first-order differential equation $\frac{\mathit{dy}}{\mathit{dt}}=\mathit{ay}$ with the initial condition $$y(0)=5$$.

Specify the initial condition as the second input to dsolve by using the == operator. Specifying condition eliminates arbitrary constants, such as C1, C2, ..., from the solution.

syms y(t) a
eqn = diff(y,t) == a*y;
cond = y(0) == 5;
ySol(t) = dsolve(eqn,cond)

ySol(t) = $$5\hspace{0.17em}{\mathrm{e}}^{a\hspace{0.17em}t}$$

Next, solve the second-order differential equation $\frac{{\mathit{d}}^2\mathit{y}}{{\mathit{dt}}^2}={\mathit{a}}^2\mathit{y}$ with the initial conditions $$y(0)=b$$ and $$y^{\prime }(0)=1$$.

Specify the second initial condition by assigning diff(y,t) to Dy and then using Dy(0) == 1.

syms y(t) a b
eqn = diff(y,t,2) == a^2*y;
Dy = diff(y,t);
cond = [y(0)==b, Dy(0)==1];
ySol(t) = dsolve(eqn,cond)

ySol(t) = 
$$\frac{{\mathrm{e}}^{a\hspace{0.17em}t}\hspace{0.17em}\left(a\hspace{0.17em}b+1\right)}{2\hspace{0.17em}a}+\frac{{\mathrm{e}}^{-a\hspace{0.17em}t}\hspace{0.17em}\left(a\hspace{0.17em}b-1\right)}{2\hspace{0.17em}a}$$

This second-order differential equation has two specified conditions, so constants are eliminated from the solution. In general, to eliminate constants from the solution, the number of conditions must equal the order of the equation.

Solve System of Differential Equations

Solve the system of differential equations

$$\begin{array}{l}\frac{\mathit{dy}}{\mathit{dt}}=\mathit{z}\\ \frac{\mathit{dz}}{\mathit{dt}}=-\mathit{y}.\end{array}$$

Specify the system of equations as a vector. dsolve returns a structure containing the solutions.

syms y(t) z(t)
eqns = [diff(y,t) == z, diff(z,t) == -y];
S = dsolve(eqns)

S = struct with fields:
    z: C2*cos(t) - C1*sin(t)
    y: C1*cos(t) + C2*sin(t)

Access the solutions by addressing the elements of the structure.

ySol(t) = S.y

ySol(t) = $$C_1\hspace{0.17em}\cos \left(t\right)+C_2\hspace{0.17em}\sin \left(t\right)$$

zSol(t) = S.z

zSol(t) = $$C_2\hspace{0.17em}\cos \left(t\right)-C_1\hspace{0.17em}\sin \left(t\right)$$

Assign Outputs to Functions or Variables

When solving for multiple functions, dsolve returns a structure by default. Alternatively, you can assign solutions to functions or variables directly by explicitly specifying the outputs as a vector. dsolve sorts the outputs in alphabetical order using symvar.

Solve a system of differential equations and assign the outputs to functions.

syms y(t) z(t)
eqns = [diff(y,t)==z, diff(z,t)==-y];
[ySol(t),zSol(t)] = dsolve(eqns)

ySol(t) = $$C_1\hspace{0.17em}\cos \left(t\right)+C_2\hspace{0.17em}\sin \left(t\right)$$

zSol(t) = $$C_2\hspace{0.17em}\cos \left(t\right)-C_1\hspace{0.17em}\sin \left(t\right)$$

Find Explicit and Implicit Solutions of Differential Equation

Solve the differential equation $$\frac{\partial }{\partial t}y(t)=e^{-y(t)}+y(t)$$. dsolve returns an explicit solution in terms of a Lambert W function that has a constant value.

syms y(t)
eqn = diff(y) == y+exp(-y)

eqn(t) = 
$$\frac{\partial }{\partial t}\mathrm{ }y\left(t\right)={\mathrm{e}}^{-y\left(t\right)}+y\left(t\right)$$

sol = dsolve(eqn)

sol = $${\mathrm{W}\text{<a href="matlab:doc symbolic/lambertw">lambertw</a>}}_0\left(-1\right)$$

To return implicit solutions of the differential equation, set the Implicit option to true. An implicit solution has the form $$F(y(t))=g(t)$$.

sol = dsolve(eqn,Implicit=true)

sol = 
$$\left(\begin{array}{c}\left({\int \frac{{\mathrm{e}}^y}{y\hspace{0.17em}{\mathrm{e}}^y+1}\mathrm{ }\mathrm{d}y|}_{y=y\left(t\right)}\right)=C_1+t\\ {\mathrm{e}}^{-y\left(t\right)}\hspace{0.17em}\left({\mathrm{e}}^{y\left(t\right)}\hspace{0.17em}y\left(t\right)+1\right)=0\end{array}\right)$$

Find Implicit Solution When No Explicit Solution Is Found

If dsolve cannot find an explicit solution of a differential equation analytically, then it returns an empty symbolic array. You can solve the differential equation by using MATLAB® numerical solver, such as ode45. For more information, see Solve a Second-Order Differential Equation Numerically.

syms y(x)
eqn = diff(y) == (x-exp(-x))/(y(x)+exp(y(x)));
S = dsolve(eqn)

Warning: Unable to find symbolic solution.

 
S =
 
[ empty sym ]
 

Alternatively, you can try finding an implicit solution of the differential equation by specifying the Implicit option to true. An implicit solution has the form $$F(y(x))=g(x)$$.

S = dsolve(eqn,Implicit=true)

S = 
$${\mathrm{e}}^{y\left(x\right)}+\frac{{y\left(x\right)}^2}{2}=C_1+{\mathrm{e}}^{-x}+\frac{x^2}{2}$$

Include Special Cases by Turning Off Internal Simplifications

Solve the differential equation $\frac{\mathit{dy}}{\mathit{dt}}=\frac{\mathit{a}}{\sqrt{\mathit{y}}}+\mathit{y}$ with condition $$y(a)=1$$. By default, dsolve applies simplifications that are not generally correct, but produce simpler solutions. For more details, see Algorithms.

syms a y(t)
eqn = diff(y) == a/sqrt(y) + y;
cond = y(a) == 1;
ySimplified = dsolve(eqn,cond)

ySimplified = 
$${\left({\mathrm{e}}^{\frac{3\hspace{0.17em}t}{2}-\frac{3\hspace{0.17em}a}{2}+\log \left(a+1\right)}-a\right)}^{2/3}$$

To return the solutions that include all possible values of the parameter $$a$$, turn off simplifications by setting IgnoreAnalyticConstraints to false.

yNotSimplified = dsolve(eqn,cond,IgnoreAnalyticConstraints=false)

yNotSimplified = 
$$\begin{array}{l}\{\begin{array}{cl}\{\begin{array}{cl}\left\{{\sigma }_1\right\}& \text{ if }-\frac{\pi }{2}<{\sigma }_2\\ \left\{{\sigma }_1,-{\left(-a+{\mathrm{e}}^{\frac{3\hspace{0.17em}t}{2}-\frac{3\hspace{0.17em}a}{2}+\log \left(a+{\left(-\frac{1}{2}+{\sigma }_3\right)}^{3/2}\right)+2\hspace{0.17em}\pi \hspace{0.17em}{C}_2\hspace{0.17em}\mathrm{i}}\right)}^{2/3}\hspace{0.17em}\left(\frac{1}{2}+{\sigma }_3\right)\right\}& \text{ if }{\sigma }_2\le -\frac{\pi }{2}\end{array}& \text{ if }{C}_2\in \mathbb{Z}\\ \varnothing & \text{ if }{C}_2\notin \mathbb{Z}\end{array}\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_1={\left(-a+{\mathrm{e}}^{\frac{3\hspace{0.17em}t}{2}-\frac{3\hspace{0.17em}a}{2}+\log \left(a+1\right)+2\hspace{0.17em}\pi \hspace{0.17em}{C}_2\hspace{0.17em}\mathrm{i}}\right)}^{2/3}\\ \\ \mathrm{ }{\sigma }_2=\text{angle}\left({\mathrm{e}}^{\frac{3\hspace{0.17em}{C}_1}{2}+\frac{3\hspace{0.17em}t}{2}}-a\right)\\ \\ \mathrm{ }{\sigma }_3=\frac{\sqrt{3}\hspace{0.17em}\mathrm{i}}{2}\end{array}$$

Find Series Solution of Differential Equation

Solve the second-order differential equation $$(x^2-1{)}^2\frac{{\partial }^2}{\partial x^2}y(x)+(x+1)\frac{\partial }{\partial x}y(x)-y(x)=0$$. dsolve returns a solution that contains a term with unevaluated integral.

syms y(x)
eqn = (x^2-1)^2*diff(y,2) + (x+1)*diff(y) - y == 0;
S = dsolve(eqn)

S = 
$$C_2\hspace{0.17em}\left(x+1\right)+C_1\hspace{0.17em}\left(x+1\right)\hspace{0.17em}\int \frac{{\mathrm{e}}^{\frac{1}{2\hspace{0.17em}\left(x-1\right)}}\hspace{0.17em}{\left(1-x\right)}^{1/4}}{{\left(x+1\right)}^{9/4}}\mathrm{ }\mathrm{d}x$$

To return series solutions of the differential equation around $$x=-1$$, set the ExpansionPoint to -1. dsolve returns two linearly independent solutions in terms of a Puiseux series expansion.

S = dsolve(eqn,ExpansionPoint=-1)

S = 
$$\left(\begin{array}{c}\frac{1}{{\left(x+1\right)}^{1/4}}-\frac{5\hspace{0.17em}{\left(x+1\right)}^{3/4}}{4}+\frac{5\hspace{0.17em}{\left(x+1\right)}^{7/4}}{48}+\frac{5\hspace{0.17em}{\left(x+1\right)}^{11/4}}{336}+\frac{115\hspace{0.17em}{\left(x+1\right)}^{15/4}}{33792}+\frac{169\hspace{0.17em}{\left(x+1\right)}^{19/4}}{184320}\\ x+1\end{array}\right)$$

Find other series solutions around the expansion point $$\infty$$ by setting ExpansionPoint to Inf.

S = dsolve(eqn,ExpansionPoint=Inf)

S = 
$$\left(\begin{array}{c}\frac{1}{6\hspace{0.17em}{x}^2}+\frac{1}{8\hspace{0.17em}{x}^4}+\frac{1}{90\hspace{0.17em}{x}^5}+1\\ x-\frac{1}{6\hspace{0.17em}{x}^2}-\frac{1}{8\hspace{0.17em}{x}^4}\end{array}\right)$$

The default truncation order of the series expansion is 6. To obtain more terms in the Puiseux series solutions, set Order to 8.

S = dsolve(eqn,ExpansionPoint=Inf,Order=8)

S = 
$$\left(\begin{array}{c}\frac{1}{6\hspace{0.17em}{x}^2}+\frac{1}{8\hspace{0.17em}{x}^4}+\frac{1}{90\hspace{0.17em}{x}^5}+\frac{37}{336\hspace{0.17em}{x}^6}+\frac{37}{1680\hspace{0.17em}{x}^7}+1\\ x-\frac{1}{6\hspace{0.17em}{x}^2}-\frac{1}{8\hspace{0.17em}{x}^4}-\frac{1}{90\hspace{0.17em}{x}^5}-\frac{37}{336\hspace{0.17em}{x}^6}\end{array}\right)$$

Specify Initial Condition for Function with Different One-Sided Limits

Solve the differential equation $\frac{\mathit{dy}}{\mathit{dx}}=\frac{1}{{\mathit{x}}^2}{\mathit{e}}^{-\frac{1}{\mathit{x}}}$ without specifying the initial condition.

syms y(x)
eqn = diff(y) == exp(-1/x)/x^2;
ySol(x) = dsolve(eqn)

ySol(x) = 
$$C_1+{\mathrm{e}}^{-\frac{1}{x}}$$

To eliminate constants from the solution, specify the initial condition $\mathit{y}\left(0\right)=1$.

cond = y(0) == 1;
S = dsolve(eqn,cond)

S = 
$${\mathrm{e}}^{-\frac{1}{x}}+1$$

The function ${\mathit{e}}^{-\frac{1}{\mathit{x}}}$ in the solution ySol(x) has different one-sided limits at $$x=0$$. The function has a right-side limit, $\underset{\mathit{x}\to 0^{+}}{\lim }{\mathit{e}}^{-\frac{1}{\mathit{x}}}=0$, but it has undefined left-side limit, $\underset{\mathit{x}\to 0^{-}}{\lim }{\mathit{e}}^{-\frac{1}{\mathit{x}}}=\infty$.

When you specify the condition y(x0) for a function with different one-sided limits at x0, dsolve treats the condition as a limit from the right, $\lim \mathit{x}\to {\mathit{x}}_0^{+}$.

Input Arguments

eqn — Differential equation or system of equations

symbolic equation | vector of symbolic equations

Differential equation or system of equations, specified as a symbolic equation or a vector of symbolic equations. Specify a differential equation by using the == operator. If eqn is a symbolic expression (without the right side), the solver assumes that the right side is 0, and solves the equation eqn == 0.

In the equation, represent differentiation by using diff. For example, diff(y,x) differentiates the symbolic function y(x) with respect to x. Create the symbolic function y(x) by using syms and solve the equation d2y(x)/dx2 = x*y(x) using dsolve.

syms y(x)
S = dsolve(diff(y,x,2) == x*y)

Specify a system of differential equations by using a vector of equations, as in syms y(t) z(t); S = dsolve([diff(y,t) == z, diff(z,t) == -y]). Here, y and z must be symbolic functions that depend on symbolic variables, which are t in this case. The right side must be symbolic expressions that depend on t, y and z. Note that Symbolic Math Toolbox™ currently does not support composite symbolic functions, that is, symbolic functions that depend on another symbolic functions.

cond — Initial or boundary condition

symbolic equation | vector of symbolic equations

Initial or boundary condition, specified as a symbolic equation or vector of symbolic equations.

When a condition contains a derivative, represent the derivative with diff. Assign the diff call to a variable and use the variable to specify the condition. For example, see Solve Differential Equations with Conditions.

Specify multiple conditions by using a vector of equations. If the number of conditions is less than the number of dependent variables, the solutions contain the arbitrary constants C1, C2,....

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: IgnoreAnalyticConstraints=false does not apply internal simplifications.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

ExpansionPoint — Expansion point of Puiseux series solution

0 (default) | number | symbolic number | symbolic variable | symbolic function | symbolic expression

Expansion point of a Puiseux series solution, specified as a number, or a symbolic number, variable, function, or expression. Specifying this option returns the solution of a differential equation in terms of a Puiseux series (a power series that allows negative and fractional exponents). The expansion point cannot depend on the series variable. For example, see Find Series Solution of Differential Equation.

Data Types: single | double | sym
Complex Number Support: Yes

IgnoreAnalyticConstraints — Option to use internal simplifications

true (default) | false

Option to use internal simplifications, specified as true or false.

By default, the solver applies simplifications while solving the differential equation, which could lead to results not generally valid. In other words, this option applies mathematical identities that are convenient, but the results might not hold for all possible values of the variables. Therefore, by default, the solver does not guarantee the completeness of results. If IgnoreAnalyticConstraints is true, always verify results returned by the dsolve function. For more details, see Algorithms.

To solve ordinary differential equations without these simplifications, set IgnoreAnalyticConstraints to false. Results obtained with IgnoreAnalyticConstraints set to false are correct for all values of the arguments. For certain equations, dsolve might not return an explicit solution if you set IgnoreAnalyticConstraints to false.

Implicit — Option to return implicit solution

false (default) | true

Option to return an implicit solution, specified as false or true. For a differential equation with variables x and y(x), an implicit solution has the form F(y(x)) = g(x).

By default, the solver tries to find an explicit solution y(x) = f(x) analytically when solving a differential equation. If dsolve cannot find an explicit solution, then you can try finding a solution in implicit form by specifying the Implicit option to true.

MaxDegree — Maximum degree of polynomial equation for which solver uses explicit formulas

2 (default) | positive integer smaller than 5

Maximum degree of polynomial equations for which the solver uses explicit formulas, specified as a positive integer smaller than 5. dsolve does not use explicit formulas when solving polynomial equations of degrees larger than MaxDegree.

Order — Truncation order of Puiseux series solution

0 (default) | positive integer | symbolic positive integer

Truncation order of a Puiseux series solution, specified as a positive integer or a symbolic positive integer. Specifying this option returns the solution of a differential equation in terms of a Puiseux series (a power series that allow negative and fractional exponents). The truncation order n is the exponent in the O-term: $$O({\mathrm{var}}^n)$$ or $$O({\mathrm{var}}^{-n})$$.

Output Arguments

S — Solutions of differential equation

symbolic expression | vector of symbolic expressions

Solutions of differential equation, returned as a symbolic expression or a vector of symbolic expressions. The size of S is the number of solutions.

y1,...,yN — Variables storing solutions of differential equations

vector of symbolic variables

Variables storing solutions of differential equations, returned as a vector of symbolic variables. The number of output variables must equal the number of dependent variables in a system of equations. dsolve sorts the dependent variables alphabetically, and then assigns the solutions for the variables to output variables or symbolic arrays.

Tips

• If dsolve cannot find an explicit or implicit solution, then it issues a warning and returns the empty sym. In this case, try to find a numeric solution using the MATLAB^® ode23 or ode45 function. Sometimes, the output is an equivalent lower-order differential equation or an integral.

• dsolve does not always return complete solutions even if IgnoreAnalyticConstraints is false.

• If dsolve returns a function that has different one-sided limits at x0 and you specify the condition y(x0), then dsolve treats the condition as a limit from the right, $$\lim x\to x_0^{+}$$.

Algorithms

If you do not set IgnoreAnalyticConstraints to false, then dsolve applies some of these rules while solving the equation:

• log(a) + log(b) = log(a·b) for all values of a and b. In particular, the following equality is applied for all values of a, b, and c:

    (a·b)^c = a^c·b^c.

• log(a^b) = b·log(a) for all values of a and b. In particular, the following equality is applied for all values of a, b, and c:

    (a^b)^c = a^b·c.

• If f and g are standard mathematical functions and f(g(x)) = x for all small positive numbers, f(g(x)) = x is assumed to be valid for all complex x. In particular:

  • log(e^x) = x

  • asin(sin(x)) = x, acos(cos(x)) = x, atan(tan(x)) = x

  • asinh(sinh(x)) = x, acosh(cosh(x)) = x, atanh(tanh(x)) = x

  • W_k(x·e^x) = x for all branch indices k of the Lambert W function.

• The solver can multiply both sides of an equation by any expression except 0.

• The solutions of polynomial equations must be complete.

Version History

Introduced before R2006a

R2020a: Return solutions of differential equations in implicit form or truncated series expansion

With dsolve:

• Use the Implicit name-value argument to return implicit solutions of differential equations.

• Use the ExpansionPoint and Order name-value arguments to specify the expansion point and truncation order of Puiseux series solutions of differential equations.

R2019b: dsolve will no longer support character vector or string inputs

dsolve will not accept equations as strings or character vectors in a future release. Instead, use symbolic expressions or sym objects to define differential equations. For example, replace inputs such as dsolve('Dy = exp(y)') with syms y(t); dsolve(diff(y,t) == exp(y)).