## 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.

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

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.

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

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)```