## 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.

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

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 |
---|---|

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

$$2{x}^{2}\frac{{d}^{2}y}{d{x}^{2}}+3x\frac{dy}{dx}-y=0.$$ | 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. $$\frac{{d}^{2}y}{d{x}^{2}}=xy(x).$$ | syms y(x) ode = diff(y,x,2) == x*y; ySol(x) = dsolve(ode) ySol(x) = C1*airy(0,x) + C2*airy(2,x) |