# Exponential Forcing Differential Equation

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1. Apr 9, 2017

### Alettix

1. The problem statement, all variables and given/known data
Solve $\frac{d^2y}{dt^2} + \omega^2y = 2te^{-t}$
and find the amplitude of the resulting oscillation when $t \rightarrow \infty$ given $y=dy/dt=0$ at $t=0$.

2. Relevant equations

3. The attempt at a solution
I have found the homogenious solution to be:
$y_h = A\cos\omega t + B\sin\omega t$
where A and B are constants.
When looking for the particular integral I tried the obvious choice $y_p = Cte^{-t}$. However, unless I have done a misstake this yields an equation system:
$(1+\omega^2)Cte^{-t} = 2te^{-t}$
$-2e^{-t}=0$
which lacks solution. Any ideas what more I should try?

I can see that as $t\rightarrow \infty$ the forcing term will tend to $0$ and hence the final amplitude should be $\sqrt{A^2+B^2}$, but I would like to find the solution to the equation..

Many Thanks! :)

2. Apr 9, 2017

### Alettix

Don't bother, I realised that $y_p = Cte^{-t}+De^{-t}$ works! :)