Differential Equations with Discontinuous Forcing Functions

[_N3WTON_]

Homework Statement

Solve the given initial-value problem.
$y'' = 1 - u(t-1)$
$y(0) = 0$
$y'(0) = 0$

Homework Equations

The Attempt at a Solution

First I took the Laplace transform of both sides:
$\mathcal{L}(y'') = \mathcal{L}(1 - u(t-1))$
$s^{2}Y(s) - sy(0) - y'(0) = \mathcal{L}(1) - \mathcal{L}(u(t-1))$
$s^{2}Y(s) = \frac{1-e^{s}}{s}$
$s^{2}Y(s) = (1-e^{s})\frac{1}{s}$
$Y(s) = (1-e^{s})\frac{1}{s^{3}}$
At this point I am sort of stuck, the solution given in the back of the book is : $\frac{1}{2}t^{2} - \frac{1}{2}u(t-1)(t-1)^{2}$
I'm having a hard time seeing how my work is going to end up as the solution given, so I am thinking maybe I didn't do something right here..

 

[_N3WTON_]

I think I may have figured out what I was doing wrong, I forgot to factor my answer...I'll post a solution momentarily...

 

[_N3WTON_]

Ok, so did figure out what I was doing wrong...I'm sorry if I've wasted anyone's time