# Non-linear second order ODE

1. Feb 10, 2013

### en51nm

Hi all,

I have a nonlinear ODE in the following form:

a x'' + b |x'|x' + c x' + d x = y

where x and y are functions of time and a,b,c and d are constants. As far as I can tell the only way to solve this is numerically, something I've managed to do successfully using a Rung-Kutta scheme. This however is a lengthy calculation, since the driving function (y) is oscillatory I am mostly interested in the phase and magnitude of the response I am looking for an alternative method.

I'm wondering if I can use a Fourier series to represent the driving function y and then calculate the resulting Fourier series for x (actually it is the x' term I am most interested in) using an FFT? But I'm not sure if this is possible due to the nonlinear term in x'.

Any help/suggestions would be greatly appreciated.

2. Feb 14, 2013

### Someone2841

This is more of a question than an answer. If x and y are both functions of time (i.e., x(t) and y(t)), then can we not pick any x(t) to solve for y(t)? For example, if x(t)=t^3 then:

$x'(t) = 3t^2, x''(t) = 6t$

Then let $y(t) = t^4 + d t^3 + c t^2 + 6at + 9b$, and you have an x(t) and y(t) such that $a x''(t) + b |x'(t)|x'(t) + c x'(t) + d x(t) = y(t)$, but there are in fact an infinite number of (x(t),y(t)) pairs, so this solution is somewhat trivial.

Might you mean x is a function of y? Then: $a x''(y) + b |x'(y)|x'(y) + c x'(y) + d x(y) = y$

Or I'm just not understanding something. Let me know.

3. Feb 14, 2013

### D H

Staff Emeritus
A driven spring mass damper system with linear and quadratic damping? Ouch.

That quadratic drag (worse than quadratic, there's an absolute value) is going to make a Fourier decomposition tough.

The damping is going to make this a stiff system, and the quadratic damping is going to make the stiffness rather interesting.

The key problem with a fixed step size integrator such as basic Runge Kutta is that the stiffness mandates that the step size be very small. It's going to take a long time to solve and will lose accuracy in the process.

Have you tried using adaptive techniques such as Runge Kutta Fehlberg, Cash-Karp, adaptive Adams Bashforth Moulton, Gauss Jackson?

Have you tried using adaptive integration techniques such as Kaps-Rentrop, various descendants of the Gear technique that are aimed specifically at stiff problems?