# Harmonic oscillator with slight non-linearity

1. Jan 5, 2012

### groinsmash

I have an interesting problem I have come across in my research. It results in the differential equation as follows:

$x''+2γ(x')^\nu+\omega_{o}^2x=g(t)$

Primes indicate the derivative with respect to $t$. $\gamma$ and $\omega$ are constants. The non-linearity comes from the first derivative $x'$ which is raised to the power of $\nu$. $\nu$ is known to be 0.12 but can be between 0 and 1. The cases where $\nu=0$ or $\nu=1$ are easy enough. But how to go about tackling an arbitrary $\nu$?

The problem may be made easier by noting that $g(t)=1$ for $t\geq0$ and $0$ for $t<0$.

Any ideas on how to go about solving it? Numerically or analytically (which would be amazing).

Thanks!

2. Jan 6, 2012

### jackmell

You gotta' get Mathematica. Here's some Mathematica code:

Code (Text):

\[Gamma] = 0.25
\[Nu] = 0.5;
\[Omega] = 0.1
mysol = NDSolve[{Derivative[2][x][t] +
2*\[Gamma]*Derivative[1][x][t]^\[Nu] + \[Omega]*x[t] ==
1, x[0] == 1, Derivative[1][x][0] == 1}, x,
{t, 0, 5}]
Plot[x[t] /. mysol, {t, 0, 5}]