[Math] Solving differential equation with the Dirac Delta Function

ordinary differential equations

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!

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