Help with differential equation

[Fiorella]

Solve with convolution integral:

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

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


Any help appreciated!

 

[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...

 

[Fiorella]

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...

Oooh I thought about that...this helps!

Thanks a lot!