Laplace DE Problem: Proving Laplace Delta(t-2)

[Hiche]

Homework Statement

Homework Equations

Laplace's transforms.

The Attempt at a Solution

Okay, so applying the Laplace on both sides yields:

$s^2Y(s) - sy(0) - y'(0) + 2sY(s) - 2y(0) + 10Y(s) = ? + 13 / s - 1$

Is $e^{-2s} / s$ the Laplace $\delta(t - 2)$? Our instructor gave us the result of the Lapace but he did not prove it. The only thing he gave us was that the Laplace of $f(t - a)\delta(t - a) = e^{-as}F(s)$. Can someone point me on how to prove this? It seems our instructor told us that we need to know the proof without him giving it to us. I know that this unit step function is defined to be 0 when 0 <= t < 2 and 1 when t >= 2.

 

[vela]

Are you sure $\delta(t)$ is the unit step function? It typically denotes the Dirac delta function.

 

[Hiche]

Yes yes. I mixed up the function. Sorry about that.

So is the Laplace of $\delta(t - 2)$ typically $e^{-2s} / s$?

 

[vela]

Which function are you talking about? If it's the delta function, then no, that's not correct. If it's the unit step, then that's right.

 

[Hiche]

The delta function. Then is it simply $e^{-2s}$?

If so, how to prove it starting with $\int^\infty_0 e^{-st}\delta(t - a)dt = e^{as}$? The proof is apparently required for our exam yet our instructor failed to provide the solution.

 

[vela]

What's the defining property of the Dirac delta function?

 

[Hiche]

..that the $\int^{a+e}_{a-e} f(t)\delta(t - a)dt = f(a)$ for $e > 0$?

I appreciate your patience but I'm relatively 'new' to this concept.

 

[vela]

Yup. Just apply that to the Laplace transform integral you have.