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.