Here's the questions and the graph
I've been struggling with this since Thursday and this is due today. I need help on problems a and b.
For a, the question is: "Write $f(t) = \sum_{n=0}^\infty \theta(t-a_n)f_n(t-a_n)$."
I think I finally made some headway.
After writing some piecewise functions: $$f(t) = \begin{cases} 1-\frac 2\pi &\text{if}& 0 < t < \pi \\ -1 + \frac 2\pi &\text{if}& \pi < t < 2\pi\\ 1-\frac2\pi(t-2\pi) &\text{if}& 2\pi < t < 3\pi\\ -1 + \frac2\pi (t-3\pi) &\text{if}& 3\pi < t <4\pi\\ \end{cases}$$
I got
$$f(t) = 1- \frac2\pi t + \sum_{n=0}^\infty \theta(t-\pi)\frac4\pi (t-\pi)$$
(where $\theta$ is the U - for unit function - in the original question.)
But should there be a $k\pi$ somewhere, or a $n\pi$ or something? How far am I from completing part a correctly?
Any guidance on parts
b: Use the Laplace transforms to solve the I.V.P. $y''+4y'+5y=f(t), y(0)=0, y'(0)=0.$ Write $Y(t)=\sum_{n=0}^\infty \theta(t-a_1)Y_n(t-a_n)$
c: Graph $Y(t)$ for $0<t<4\pi$.
or d: Predict $\lim {t>\infty}|Y(t)|$.
would be appreciated as well. This is due in about 9 hours and is a take-home quiz for a differential equations class where we're allowed to use any resources we have (short of paying) to solve the problem.
Thank you so much for any help!