Solve Differential Equation

Solve a differential equation analytically by using the dsolve function, with or without initial conditions. To solve a system of differential equations, see Solve a System of Differential Equations.

First-Order Linear ODE

Solve this differential equation.

$$\frac{\mathit{dy}}{\mathit{dt}}=\mathit{ty}.$$

First, represent $$y$$ by using syms to create the symbolic function y(t).

syms y(t)

Define the equation using == and represent differentiation using the diff function.

ode = diff(y,t) == t*y

ode(t) = 
$$\frac{\partial }{\partial t}\mathrm{ }y\left(t\right)=t\hspace{0.17em}y\left(t\right)$$

Solve the equation using dsolve.

ySol(t) = dsolve(ode)

ySol(t) = 
$$C_1\hspace{0.17em}{\mathrm{e}}^{\frac{t^2}{2}}$$

Solve Differential Equation with Condition

In the previous solution, the constant $$C_1$$ appears because no condition was specified. Solve the equation with the initial condition y(0) == 2. The dsolve function finds a value of $$C_1$$ that satisfies the condition.

cond = y(0) == 2;
ySol(t) = dsolve(ode,cond)

ySol(t) = 
$$2\hspace{0.17em}{\mathrm{e}}^{\frac{t^2}{2}}$$

If dsolve cannot solve your equation, then try solving the equation numerically. See Solve a Second-Order Differential Equation Numerically.

Nonlinear Differential Equation with Initial Condition

Solve this nonlinear differential equation with an initial condition. The equation has multiple solutions.

$$\begin{array}{l}{\left(\frac{\mathit{dy}}{\mathit{dt}}+\mathit{y}\right)}^2=1,\\ \mathit{y}\left(0\right)=0.\end{array}$$

syms y(t)
ode = (diff(y,t)+y)^2 == 1;
cond = y(0) == 0;
ySol(t) = dsolve(ode,cond)

ySol(t) = 
$$\left(\begin{array}{c}{\mathrm{e}}^{-t}-1\\ 1-{\mathrm{e}}^{-t}\end{array}\right)$$

Second-Order ODE with Initial Conditions

Solve this second-order differential equation with two initial conditions.

$$\begin{array}{l}\frac{{\mathit{d}}^2\mathit{y}}{{\mathit{dx}}^2}=\cos \left(2\mathit{x}\right)-\mathit{y},\\ \mathit{y}\left(0\right)=1,\\ {\mathit{y}}^{\prime }\left(0\right)=0.\end{array}$$

Define the equation and conditions. The second initial condition involves the first derivative of y. Represent the derivative by creating the symbolic function Dy = diff(y) and then define the condition using Dy(0)==0.

syms y(x)
Dy = diff(y);

ode = diff(y,x,2) == cos(2*x)-y;
cond1 = y(0) == 1;
cond2 = Dy(0) == 0;

Solve ode for y. Simplify the solution using the simplify function.

conds = [cond1 cond2];
ySol(x) = dsolve(ode,conds);
ySol = simplify(ySol)

ySol(x) = 
$$1-\frac{8\hspace{0.17em}{\sin \left(\frac{x}{2}\right)}^4}{3}$$

Third-Order ODE with Initial Conditions

Solve this third-order differential equation with three initial conditions.

$$\begin{array}{l}\frac{{\mathit{d}}^3\mathit{u}}{{\mathit{dx}}^3}=\mathit{u},\\ \mathit{u}\left(0\right)=1,\\ {\mathit{u}}^{\prime }\left(0\right)=-1,\\ {{\mathit{u}}^{\prime }}^{\prime }\left(0\right)=\pi .\end{array}$$

Because the initial conditions contain the first- and second-order derivatives, create two symbolic functions, Du = diff(u,x) and D2u = diff(u,x,2), to specify the initial conditions.

syms u(x)
Du = diff(u,x);
D2u = diff(u,x,2);

Create the equation and initial conditions, and solve it.

ode = diff(u,x,3) == u;
cond1 = u(0) == 1;
cond2 = Du(0) == -1;
cond3 = D2u(0) == pi;
conds = [cond1 cond2 cond3];

uSol(x) = dsolve(ode,conds)

uSol(x) = 
$$\frac{\pi \hspace{0.17em}{\mathrm{e}}^x}{3}-{\mathrm{e}}^{-\frac{x}{2}}\hspace{0.17em}\cos \left(\frac{\sqrt{3}\hspace{0.17em}x}{2}\right)\hspace{0.17em}\left(\frac{\pi }{3}-1\right)-\frac{\sqrt{3}\hspace{0.17em}{\mathrm{e}}^{-\frac{x}{2}}\hspace{0.17em}\sin \left(\frac{\sqrt{3}\hspace{0.17em}x}{2}\right)\hspace{0.17em}\left(\pi +1\right)}{3}$$

More ODE Examples

This table shows examples of differential equations and their Symbolic Math Toolbox™ syntax.

Differential Equation	Commands Using Symbolic Math Toolbox
$$\begin{array}{l}\frac{\mathit{dy}}{\mathit{dt}}+4\mathit{y}\left(\mathit{t}\right)={\mathit{e}}^{-\mathit{t}},\\ \mathit{y}\left(0\right)=1.\end{array}$$	>> syms y(t) >> ode = diff(y)+4*y == exp(-t); >> cond = y(0) == 1; >> ySol(t) = dsolve(ode,cond) ySol(t) = exp(-t)/3 + (2*exp(-4*t))/3
$$2{\mathit{x}}^2\frac{{\mathit{d}}^2\mathit{y}}{{\mathit{dx}}^2}+3\mathit{x}\frac{\mathit{dy}}{\mathit{dx}}-\mathit{y}=0.$$	>> syms y(x) >> ode = 2*x^2*diff(y,x,2)+3*x*diff(y,x)-y == 0; >> ySol(x) = dsolve(ode) ySol(x) = C1/(3*x) + C2*x^(1/2)
The Airy equation. $$\frac{{\mathit{d}}^2\mathit{y}}{{\mathit{dx}}^2}=\mathit{x}\mathit{y}\left(\mathit{x}\right).$$	>> syms y(x) >> ode = diff(y,x,2) == x*y; >> ySol(x) = dsolve(ode) ySol(x) = C1*airy(0,x) + C2*airy(2,x)
Puiseux series solution. $$\left({\mathit{x}}^2+1\right)\frac{{\mathit{d}}^2\mathit{y}}{{\mathit{dx}}^2}-2\mathit{x}\frac{\mathit{dy}}{\mathit{dx}}+\mathit{y}=0.$$	>> syms y(x) a >> ode = (x^2+1)*diff(y,x,2)-2*x*diff(y,x)+y == 0; >> Dy = diff(y,x); >> cond = [Dy(0) == a; y(0) == 5]; >> ySol(x) = dsolve(ode,cond,ExpansionPoint=0) ySol(x) = - (a*x^5)/120 - (5*x^4)/24 + (a*x^3)/6 - (5*x^2)/2 + a*x + 5