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 Differential Equation with Condition

• Nonlinear Differential Equation with Initial Condition

• Second-Order ODE with Initial Conditions

• Third-Order ODE with Initial Conditions

• More ODE Examples

First-Order Linear ODE

Solve this differential equation.

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

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

Solve the equation using dsolve.

ySol(t) = dsolve(ode)

ySol(t) =
C1*exp(t^2/2)

Solve Differential Equation with Condition

In the previous solution, the constant C1 appears because no condition was specified. Solve the equation with the initial condition y(0) == 2. The dsolve function finds a value of C1 that satisfies the condition.

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

ySol(t) =
2*exp(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.

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

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

ySol(t) =
 exp(-t) - 1
 1 - exp(-t)

Second-Order ODE with Initial Conditions

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

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

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 - (8*sin(x/2)^4)/3

Third-Order ODE with Initial Conditions

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

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

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) =
 
(pi*exp(x))/3 - exp(-x/2)*cos((3^(1/2)*x)/2)*(pi/3 - 1) -...
(3^(1/2)*exp(-x/2)*sin((3^(1/2)*x)/2)*(pi + 1))/3

More ODE Examples

This table shows examples of differential equations and their Symbolic Math Toolbox™ syntax. The last example is the Airy differential equation, whose solution is called the Airy function.

Differential Equation	MATLAB® Commands
$$\begin{array}{l}\frac{dy}{dt}+4y(t)=e^{-t},\\ y\left(0\right)=1.\end{array}$$	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
$$2x^2\frac{d^{2}y}{d{x}^2}+3x\frac{dy}{dx}-y=0.$$	syms y(x) ode = 2*x^2*diff(y,x,2)+3*x*diff(y,x)-y == 0; ySol(x) = dsolve(ode) ySol(x) = C2/(3*x) + C3*x^(1/2)
The Airy equation. $$\frac{d^{2}y}{d{x}^2}=xy(x).$$	syms y(x) ode = diff(y,x,2) == x*y; ySol(x) = dsolve(ode) ySol(x) = C1*airy(0,x) + C2*airy(2,x)