Chapter 4.6, Problem 8P

(a) Find the solution of the initial value problem in Problem $6$.

(b) Identify the transient and steady-state parts of the solution.

(c) Plot the graph of the steady-state solution.

(d) If the given external force is replaced by a force A $\exp \left(iwt\right)$ of frequency $w$, find the frequency response $G\left(iw\right)$, the gain $\left|G\left(iw\right)\right|$, and the phase $\varphi \left(w\right)=-\arg \left(G\left(iw\right)\right)$. Then find the value of w for which the gain is maximum. Plot the graphs of $\left|G\left(iw\right)\right|$ and $\varphi \left(w\right)$.