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This example shows you how to convert a second-order differential equation into a system of differential equations that can be solved using the numerical solver `ode45`

of MATLAB®.

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.

Use `odeToVectorField`

to rewrite this second-order differential equation

$$\frac{{d}^{2}y}{d{t}^{2}}=(1-{y}^{2})\frac{dy}{dt}-y$$

using a change of variables. Let $$y(t)={Y}_{1}$$and $$\frac{dy}{dt}={Y}_{2}$$ such that differentiating both equations we obtain a system of first-order differential equations.

$$\begin{array}{c}\frac{d{Y}_{1}}{dt}={Y}_{2}\\ \frac{d{Y}_{2}}{dt}=-({Y}_{1}^{2}-1){Y}_{2}-{Y}_{1}\end{array}$$

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

V =$$\left(\begin{array}{c}{Y}_{2}\\ -\left({{Y}_{1}}^{2}-1\right)\hspace{0.17em}{Y}_{2}-{Y}_{1}\end{array}\right)$$

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 = `*function_handle with value:*
@(t,Y)[Y(2);-(Y(1).^2-1.0).*Y(2)-Y(1)]

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 using `linspace`

to generate 100 points in the interval [0,20] and `deval`

to evaluate the solution for each point.

fplot(@(x)deval(sol,x,1), [0, 20])