Solve a Second-Order Differential Equation Numerically

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.

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

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 = 
$$\left(\begin{array}{c}{Y}_2\\ -\left({Y_1}^2-1\right)\hspace{0.17em}{Y}_2-Y_1\end{array}\right)$$

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 = function_handle with value:
    @(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.

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