Laplace transform with unit step function

[Juanriq]

Homework Statement

I'm trying to take the laplace transfrom of t H(t) where H(t) is the unit step function. Also, in a separate problem I get e^{-t} H(t) - e^{-t}H(t-1) and I am wondering how to manipulate it properly

Homework Equations

L \{ f(t-a) H(t-a) \} = e^{as}F(s)

The Attempt at a Solution

For the first part, I thought that L \{ t H(t) \} should just give \frac{1}{s^2} back out, but the answer key in the book I'm using says that it is just \frac{1}{s}.



For the second part of my question, I know we have to manipulate the exponential, but how would I manipulate them? For instance, I want e^{t}H(t), but can't I only multiply by a constant? Obviously e^{-t}e^{2t}H(t) would do what I want, but now I'm introuducing something I can't factor outside the transform. Similarly for the second, e^{2t-1}e^{-t} [\latex] would do the trick... Any help is appreciated!

 

[vela]

Your book is wrong about L[tH(t)]. Your answer is right.

For the second problem, use the fact that e^-t=e^-(t-1)e^-1.