Struggling to Find X(t) Using Inverse Laplace Transform

[schapman22]

Homework Statement

Having found the laplace transform of a differential equation. I must now find X(t). All of my work is attached. The problem I am having is fitting my function of s to my table of transforms. I tried using partial fractions but it took me in a loop.

The Attempt at a Solution

I attached everything.

 

[schapman22]

I know the answer is e^t which means somehow 1/(s-1)² becomes 1/(s-1). But I don't understand how to get rid of the other (s-1) in the denominator.

 

[schapman22]

Hey I still can't figure this one out. If anyone has any advice it would be very much appreciated. Thank you.

 

[LCKurtz]

I have trouble reading your images, but if you are trying to inverse $\frac 1 {(s-1)^2}$ you can use one of the shifting theorems$$
\mathcal L^{-1}f(s-a) = e^{at}\mathcal L^{-1}f(s)$$Do you see how to use that? Your answer isn't just $e^t$.