Need HELP with Laplace transform

[HAL10000]

So I've been searching on the internet and in my textbook and I can't find a simple enough explanation of the Laplace transform of a product of functions. Is it absolutely necessary to evaluate the integral using different methods or is there an easier way?

 

[dantham]

If by laplace of a product of functions you mean something such as L(t sin at), then there are several methods.

(1) We could differentiate the product of functions are make use of the following laplace property:
L(f'') = s^2 L(f) - sf(0) - f'(0)

Let f(t) = t sin at
f'(t) = at cos at + sin at
f''(t) = -a^2 t sin at + a cos at

L(f''(t)) = -a^2 L(t sin at) + a L(cos at)
= s^2 L(t sin at) - s (0 sin 0) - (0 + sin 0)
= s^2 L(t sin at)

Rearranging, we get
(s^2 + a^2) L(t sin at) = aL(cos at)

Which you should be able to solve since you know L(cos at)

(2) Or you could go with the definition of the Laplace transform.
First start with L(sin at) = a/ (a^2 + s^2) = integrate (e^-st . (sin at)) dt
differentiate both the Left hand side, and right hand side, with respect to s

differentiate (a/ (a^2 + s^2)) = integrate (-te^-st . (sin at)) dt = - integrate (e^-st . tsin at) dt which by definition is L(t sin at)

 

[HAL10000]

Got it thanks! :) Great explanation!