# Highly nonlinear ODE

1. Dec 4, 2015

### tburke2

I have an equation of motion given by
$$f(z(t),t) = \frac{d^2z}{dt^2} + A\frac{dz}{dt} + B$$
where
$$f(z(t),t) = [(\frac{C}{z^2+C^2})^2-(\frac{D}{z^2+D^2})^4]^2(1+cos(wt))$$
and $A,B,C,D,$ and $w$ are constants

Is it possible to solve this for $z(t)$? I have been solving it numerically using Matlab's ODE solver but as this model is used to fit a set of experimental data (constants $A$ and $B$ are varied until a reasonably small error is achieved between this and the experimental data) it would greatly reduce computational time if a solution or even a close approximation can be found.

I know from numerically solving $z(t)$ that it is periodic so there must be a Fourier series that can be used to find a solution. I have basic knowledge of Fourier analysis and since $f$ is a function of $z$ and $t$, and $z$ is dependent on $t$, I'm unsure how to do this. If someone could point me in the right direction it would be much appreciated.

2. Dec 9, 2015