I've got a differential equation to solve with the Dirac Delta Function and I'm not really sure how to handle it. I have been instructed to use the Laplace Transform as my method of solution.
Here is the equation:
$y''+8y'+41y=δ(t-\pi)+δ(t-3\pi)$, $y(0)=1, y'(0)=0$
I have no idea where to begin here. I took the Laplace transform but at this point I'm unsure exactly how to decompose the function after I solved for $y$.
Thanks for any help.
Best Answer
Taking the Laplace transform is the good way!
HINTS:
$$\mathcal{L}\left(\delta(t - \pi), s\right) = e^{-\pi s}$$
$$\mathcal{L}\left(\delta(t - 3\pi), s\right) = e^{-3\pi s}$$
Can you go on? The LT of the other terms are straightforward!