Chapter 6.4, Problem 16P

(a)

To determine

To sketch: The graph of $g\left(t\right)=u_{\frac{3}{2}}\left(t\right)-u_{\frac{5}{2}}\left(t\right)$.

(b)

To determine

The solution of the initial value problem, $u^{″}+\frac{1}{4}{u}^{\prime }+u=kg\left(t\right),u\left(0\right)=0,u^{\prime }\left(0\right)=0$.

(c)

To determine

To plot: The solution for $k=\frac{1}{2},1,2$ and to describe the principal features of the solution and to explain how the features depend on k.

(d)

To determine

To estimate: The smallest value of k for which the solution $u\left(t\right)$ reaches the value 2.

(e)

To determine

The time $\tau$ after which $\left|u\left(t\right)\right|<0.1$ for all $t>\tau$.