# Solve eqution for oscillation

1. May 8, 2013

### skrat

1. The problem statement, all variables and given/known data
Solve: $\ddot{x}+\Omega^{2} x=D+\frac{C}{2}+Ecos\omega t+\frac{C}{2}cos2\omega t$

2. Relevant equations

3. The attempt at a solution
I got a hint to use $x=\alpha sin\omega t+\beta cos\omega t$ so $\ddot{x}=-\alpha ^{2}\omega ^{2}sin\omega t-\beta ^{2}\omega ^{2}cos\omega t$ in the equation above than:
$(-\alpha ^{2}\omega ^{2}sin\omega t-\beta ^{2}\omega ^{2}cos\omega t)+\Omega^{2}x=\alpha sin\omega t+\beta cos\omega t=D+\frac{C}{2}+Ecos\omega t+\frac{C}{2}cos2\omega t$
Which gives me 4 separate equations depending on $sin\omega t$, $cos\omega t$, $cos2\omega t$ and constant:

first: $-\alpha ^{2}\omega ^{2}+\Omega ^{2}\alpha=0$
second: $-\beta ^{2}\omega ^{2}+\Omega ^{2}\beta =E$
third: $\frac{C}{2}=0$
fourth: $D+\frac{C}{2}=0$

Forth and third together say that $D=0$ and $C=0$
First says that:
$\alpha ^{2}\omega ^{2}=\Omega ^{2}\alpha$
$\alpha =(\frac{\Omega }{\omega })^{2}$
But for second I am not sure, whether I can divide it with $\beta$ (probably not since it could be equal to 0) or how do I solve it?