Problems
In each of Problems 1 through 8:
a. Sketch the graph of the forcing function on an appropriate interval.
b. Find the solution of the given initial value problem.
c. Plot the graph of the solution.
d. Explain how the graphs of the forcing function and the solution are related.
1. $y'' + y = f(t)$; $y(0) = 0$, $y'(0) = 1$; $f(t) = \begin{cases} 1, & 0 \le t < 3\pi \\ 0, & 3\pi \le t < \infty \end{cases}$
2. $y'' + 2y' + 2y = h(t)$; $y(0) = 0$, $y'(0) = 1$; $h(t) = \begin{cases} 1, & \pi \le t < 2\pi \\ 0, & 0 \le t < \pi \text{ or } t \ge 2\pi \end{cases}$
3. $y'' + 4y = \sin(t) - u_{2\pi}(t)\sin(t - 2\pi)$; $y(0) = 0$, $y'(0) = 0$
4. $y'' + 3y' + 2y = f(t)$; $y(0) = 0$, $y'(0) = 0$; $f(t) = \begin{cases} 1, & 0 \le t < 10 \\ 0, & t \ge 10 \end{cases}$