# Solve a Second-Order Differential Equation

A typical approach to solving higher-order ordinary differential equations is to convert them to systems of first-order differential equations, and then solve those systems. The example uses Symbolic Math Toolbox to convert a second-order ODE to a system of first-order ODEs. Then it uses the MATLAB® solver ode45 to solve the system.

## Contents

## Rewrite the Second-Order ODE as a System of First-Order ODEs

Use `odeToVectorField` to rewrite this second-order differential equation as a system of first-order differential equations.

```
syms y(t)
V = odeToVectorField(diff(y, 2) == (1 - y^2)*diff(y) - y)
```

V = Y[2] - (Y[1]^2 - 1)*Y[2] - Y[1]

## Generate MATLAB function

The MATLAB ODE solvers do not accept symbolic expressions as an input. therefore, before you can use a MATLAB ODE solver to solve the system, you must convert that system to a MATLAB function. Generate a MATLAB function from this system of first-order differential equations using matlabFunction with V as an input.

M = matlabFunction(V,'vars', {'t','Y'})

M = @(t,Y)[Y(2);-(Y(1).^2-1.0).*Y(2)-Y(1)]

## Solve the System of First-Order ODEs

To solve this system, call the MATLAB ode45 numerical solver using the generated MATLAB function as an input.

sol = ode45(M,[0 20],[2 0]);

## Plot the Solution

Plot the solution using linspace to generate 100 points in the interval [0,20] and deval to evaluate the solution for each point.

x = linspace(0,20,100); y = deval(sol,x,1); plot(x,y);