# Help with differential equation

1. Aug 2, 2009

### Fiorella

Solve with convolution integral:

Click http://i3.photobucket.com/albums/y62/Phio/eq.jpg" [Broken] to see the equation.

So far what I've got is http://i3.photobucket.com/albums/y62/Phio/attempt.jpg" [Broken]. I don't know what else to do from there, or if what i'm doing is right...

Any help appreciated!

Last edited by a moderator: May 4, 2017
2. Aug 3, 2009

### gabbagabbahey

First, $c_3$ should be equal to $k$ from your DE should it not?

Second,

$$X(s)=\frac{F(s)}{s^2+k}+\frac{c_1s}{s^2+k}+\frac{c_2}{s^2+k}\implies x(t)=\mathcal{L}^{-1} \left\{ \frac{F(s)}{s^2+k}+\frac{c_1s}{s^2+k}+\frac{c_2}{s^2+k} \right\}$$
$$=\mathcal{L}^{-1} \left\{ \frac{F(s)}{s^2+k} \right\}+\mathcal{L}^{-1} \left\{ \frac{c_1s}{s^2+k} \right\}+\mathcal{L}^{-1} \left\{ \frac{c_2}{s^2+k} \right\}$$

You shouldn't have much trouble doing the last two inverse Laplace transforms, and the first one can be done using the convolution rule....

3. Aug 3, 2009

### Fiorella

Oooh I thought about that...this helps!

Thanks a lot!