# Documentation

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## Solve Differential Equation

Solve a differential equations by using the dsolve function, with or without initial conditions. This page shows how to solve single differential equations. To solve a system of differential equations, see Solve a System of Differential Equations.

### First-Order Linear ODE with Initial Condition

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)

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)

### 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.

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

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.

$\begin{array}{l}\frac{{d}^{3}u}{d{x}^{3}}=u,\\ u\left(0\right)=1,\text{\hspace{0.17em}}\\ {u}^{\prime }\left(0\right)=-1,\\ \text{\hspace{0.17em}}{{u}^{\prime }}^{\prime }\left(0\right)=\pi .\end{array}$

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) =

(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

$\begin{array}{l}\frac{dy}{dt}+4y\left(t\right)={e}^{-t},\\ y\left(0\right)=1.\end{array}$

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

$2{x}^{2}\frac{{d}^{2}y}{d{x}^{2}}+3x\frac{dy}{dx}-y=0.$

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.

$\frac{{d}^{2}y}{d{x}^{2}}=xy\left(x\right).$

syms y(x)
ode = diff(y,x,2) == x*y;
ySol(x) = dsolve(ode)
ySol(x) =
C1*airy(0,x) + C2*airy(2,x)