How to Solve a Laplace Transform Problem with Dirac Delta Function?

[aero_eng]

Homework Statement

Given y''+9y=\delta(t-\pi)
y(0) = y'(0) = 1

Homework Equations

Obtain y = ...

The Attempt at a Solution

I have tried to Laplace transform the RHS and the LHS
But I am not sure how to do it. Please help!

 

[Domnu]

Here's the Wikipedia article: http://en.wikipedia.org/wiki/Laplace_transform. You can just transform each entry (y'', 9y, etc.) individually and add them together because of the linearity of the Laplace Transform. Now, the Laplace transform of delta(t-C) for some constant C is just exp(-Cs). Plug all of this in and solve for F(s). For the resulting function, take the inverse Laplace transform (which may be slightly more difficult... you may have to use convolutions, etc.)

 

[Defennder]

Here are some formulae to start you off:

$$L[y^{(n)}] = s^n L(y) - s^{n-1}y(0) - s^{n-2}y'(0) - ... - y^{(n-1)}(0)$$
$$L[\delta(t-a)] = e^{-as}$$

As for the resulting L(y) expression, you only need know the Laplace transform of cos wt, sin wt. and u(t-a)f(t-a) to solve for y in the time-domain.