# Rayleigh's differential equation

by atrus_ovis
Tags: differential, equation, rayleigh
 P: 101 In Rayleigh's DE : http://www.wolframalpha.com/input/?i...ntial+equation What does mu stand for?
 Sci Advisor HW Helper P: 11,835 It's a mere real positive parameter, just like in other famous ODE's, for example the Bessel differential equation and . http://www.wolframalpha.com/input/?i...ntial+equation
 P: 101 Well, i am asked to numerically solve it and produce a phase diagram. Should its value be given to me?
I guess it should, so you're free to choose any value you want: Take $\mu =1$ and solve it numerically.
 P: 101 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?