How to solve this differential equation of second order (non-linear)?

Question

How to solve the system using the Runge-Kutta Felberg method (differential equation of second order (non-linear))?

Answer 1

We have the system

$$\tag 1 x''+5x' + 44y' + 0.5 xy = \sin 5t \\ y'' + 3y' + 2 x' - x y^2 = \cos 10 t \\x(0) = 0.1, x'(0) = 0.02, y(0) = 0.2, y'(0) = 0.01$$

We want to solve this system using Runge-Kutta-Fehlburg (RKF), but that method is used for first order equations of the form

$$y' = f(t, y), a \le t \le b, y(a) = \alpha.$$

In order to use this method, we must convert $(1)$ into a system of first order equations first, then we can use the RKF algorithm. We proceed as follows

• $x_1 = x \implies x_1(0) = x(0) = 0.1$
• $x_1' = x' = x_2 \implies x_1'(0) = x'(0) = 0.02$
• $y_1 = y \implies y_1(0) = y(0) = 0.2$
• $y_1' = y' = y_2 \implies y_1'(0) = y'(0) = 0.01$
• $x_2' = x'' = -5x'-44y-0.5x y + \sin 5t = -5 x_2 - 44 y_1 -0.5 x_1 y_1 + \sin 5t$
• $y_2' = y'' = -3y' - 2 x' + x y^2 + \cos 10 t = -3 y_2 - 2 x_2 + x_1 y_1^2 + \cos 10 t$

Summarizing our new system

$ x_1' = x_2 \\x_2' = -5 x_2 - 44 y_1 -0.5 x_1 y_1 + \sin 5t \\y_1' = y_2\\y_2' = -3 y_2 - 2 x_2 + x_1 y_1^2 + \cos 10 t \\ x_1(0) = 0.1 , x_1'(0) = 0.02, y_1(0) = 0.2, y_1'(0) = 0.01$

Now you can apply the algorithm (with modifications to account for a first order system) and should end up with plots for each state variable that looks something like