Rayleigh's differential equation

[atrus_ovis]

In Rayleigh's DE :
http://www.wolframalpha.com/input/?i=rayleigh+differential+equation

What does mu stand for?

 

[dextercioby]

It's a mere real positive parameter, just like in other famous ODE's, for example the Bessel differential equation and <n>.

http://www.wolframalpha.com/input/?i=Bessel+differential+equation

 

[atrus_ovis]

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

 

[dextercioby]

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

 

[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
$$y_1 = y , y_1' = y_2 \\<br /> y_2' = \mu y_2 - \frac{\mu (y_2)^3}{3} - y_1$$

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'
$$k_1 = hf(x_n,y_n) = hy_2(n)\\<br /> k_2 = hf(x_n + 0.5h,y_n + 0.5k_1) = ?$$
I should numerically approximate the intermmediate values as well?