Question

Find the LaPlace Transform of f(t) = tcos(2t).

Answer 1

Laplace*

Answer 2

\[F(s)=\frac{ s^2-4 }{ (s^2+4)^2 }\]
(To evaluate the laplace transform, do integration by parts)

Answer 3

okay where do i start?

Answer 4

Do I take the integral of tcos2t?

Answer 5

Here is the identity we want to use,\[\huge \mathcal L[t^n \cdot f(t)]=(-1)^n \frac{d^n}{ds^n}F(s)\]In this problem, f(t) is cos(2t).
I'm not sure why you would use the definition. It's better to use the shortcuts on these types of problems once you've learned them :)

Answer 6

im so confused

Answer 7

\[\large \mathcal L [t\cdot \cos(2t)]=(-1)\frac{d}{ds}\mathcal L[\cos(2t)]\]

\[\large (-1)\frac{d}{ds}\left[\frac{s}{s^2+4}\right]\]And then just take the derivative from there :D

Answer 8

Confused about what? :D Have you learned the Laplace of Cosine and sine yet?
This doesn't require too much work, it's just a simple trick to get through.

Answer 9

For notation, I will write the laplace transform of f(t) as F(s) anyways:
F(s)=the integral of \[t*e ^{-st}*\Re(e^{iat})dt\] then do integration by parts on that (not hard). Or you could use that identity.

Answer 10

(it's also a definite/improper integral from 0 to infinity).

Answer 11

right

Answer 12

So identities and integration by parts are my only options?

Answer 13

Essentially

Answer 14

Okay thanks :)