Chapter 5.6, Problem 16P

A certain spring-mass system satisfies the initial value problem

$u\text{'}\text{'}+0.25u\text{'}+u=kg(t),$ $u(0)=0,$ $u\text{'}(0)=0,$

Where $g(t)$ is equal to 1 over the time interval $\frac{3}{2}<t<\frac{5}{2},$ is zero elsewhere, and $k>0$ is a parameter.

a) Sketch the graph of $g(t)$

b) Solve the initial value problem.

c) Plot the solution for $k=\frac{1}{2}$, $k=1$ and $k=2,$ describe the principle features pf the solution and how they depend on $k$.

d) Find, to two decimal places, the smallest value of $k$ for which the solution $u(t)$ reaches the value 2.

e) Suppose $k=2$. Find the time $\tau$ after which $\left|u(t)\right|<0.1$ for all $t>\tau .$