# Rayleigh's differential equation

1. Dec 29, 2012

### atrus_ovis

2. Dec 29, 2012

### dextercioby

3. Dec 29, 2012

### atrus_ovis

Well, i am asked to numerically solve it and produce a phase diagram.
Should its value be given to me?

4. Dec 29, 2012

### dextercioby

I guess it should, so you're free to choose any value you want: Take $\mu =1$ and solve it numerically.

5. Dec 30, 2012

### atrus_ovis

You're right , it was supposed to be given.
Rayleigh's DE is $y''-\mu y' + \frac{\mu (y')^3}{3} + y = 0$
By rearranging it to a system of DEs, you get
$$y_1 = y , y_1' = y_2 \\ y_2' = \mu y_2 - \frac{\mu (y_2)^3}{3} - y_1$$

So i have only the derivative of y2 , i.e. the 2nd derivative of y1.
Since i don't have an analytical description of y2 , how do i compute it with specific parameters, according to the numerical method.
For example, for the classic Runge Kutta method,where f = y'
$$k_1 = hf(x_n,y_n) = hy_2(n)\\ k_2 = hf(x_n + 0.5h,y_n + 0.5k_1) = ?$$
I should numerically approximate the intermmediate values as well?