# Inverse Laplace transform with unit step function

1. Jul 8, 2012

### shaqywacky

Hello again.

First off, I wasn't sure how to say this in the title but I'm not taking the inverse Laplace transform of a unit step function. I'm taking the Laplace transform of something that comes out to the unit step function.

I have this question, which is a similar version of the question I am trying to solve. This is also the solution but I have no idea what happened.

(sorry, I don't know why that is so small, click to make it bigger)

So the first part of that image is just the inverse Laplace transform I am trying to solve. The first step I saw was to do partial fractions. When I do that I get:
$\frac{e}{s+1} + \frac{1}{s}$
I'm omitting the inverse Laplace transform here because I don't know how to do it in latex.

But as you can see, the solution maunual got:
$\frac{e^{-s}}{s} - \frac{e^{-s}}{s+1}$

Clearly there is something I do not understand about this. I can get the correct answer after this, I just don't understand this step.

Any help would be greatly appreciated.

Thank you.

Last edited: Jul 8, 2012
2. Jul 9, 2012

### JJacquelin

Hi !
You question seems confused.
You must not use the same variable (s) for the function and for the Laplace transform of the function.
"The Laplace transform of f(x) is F(s)"
or
"The inverse Laplace transform of F(s) is f(x)"
Then, write clearly the expression of the known function : is it f(x) or F(s) ?

3. Jul 9, 2012

### shaqywacky

I don't think I did. I never took any inverse Laplace transforms in my post. I'm talking about inside the Laplace transform. I didn't know how to use the Laplace transform in Latex, so I omitted it and made a note of it but I don't think I was being very clear. I'm just omitting the notation of the Laplace transform. So the equations I posted (besides the ones in the picture) are what's inside of the Laplace transform I'm trying to solve.

To try to clarify, I just don't understand the first step that is in the picture. So I don't see how they went from
$\frac{e^{-s}}{s(s-1)}$

to

$\frac{e^{-s}}{s} - \frac{e^{-s}}{s-1}$

Thanks.

4. Jul 10, 2012

### JJacquelin

Hi !

Do you know how to write a polynomial fraction as a sum of simple terms ?
Have a look to the attachment :

File size:
7.2 KB
Views:
97