- #1
Destroxia
- 204
- 7
Homework Statement
##y'' + 2y' + 2y = h(t); y(0)=0, y'(0)=1,
h(t) = \begin{cases} t &, \pi \leq t < 2\pi \\
0 &, 3\pi \leq t < \infty \\
\end{cases}##
2. Homework Equations
The Attempt at a Solution
Take the Laplace of both sides:[/B]
##\mathcal{L}(y'' + 2y' + 2y) = \mathcal{L}(h(t)) ##
##s^2 F(s) - sf(0) - f'(0) + 2sF(s) - 2f(0) + 2F(s) = \mathcal{L}(h(t)) ##
##(s^2 + 2s +2)F(s) - 1 = \mathcal{L}(h(t)) = \int_{\pi}^{2\pi} te^{-st}##
##(s^2 + 2s +2)F(s) - 1 =\frac {\pi e^{-\pi s}} {s} + \frac {e^{-\pi s}} {s^2} - \frac {2\pi e^{-2\pi s}} {s} - \frac {e^{2\pi s}} {s^2} ##
Solve for F(s):
This is where I always screw up, If I even dare to add the 1 to the other side and divide by ##(s^2 + 2s + 2)## I get a ridiculously long expansion, that I don't even think is plausible for a practice problem, or any other problem.
I have no idea where to continue once I get to this point.