Chapter 3.7, Problem 18P

(a)

To determine

The solution of the initial value problem $m{u}^{″}+\gamma u^{\prime }+ku=0$, $u\left(0\right)=u_0,u^{\prime }\left(0\right)=v_0$ if ${\gamma }^2-4km<0$.

(b)

To determine

The solution of the initial value problem $m{u}^{″}+\gamma u^{\prime }+ku=0$, $u\left(0\right)=u_0,u^{\prime }\left(0\right)=v_0$ in the form of $u\left(t\right)={\mathrm{Re}}^{-\frac{\gamma t}{2m}}\cos \left(\mu t-\delta \right)$ if ${\gamma }^2-4km<0$ and then determine R in terms of $m,\gamma ,k,u_0\text{ and }{v}_0$.

(c)

To determine

To analyse: The values of $R=\sqrt{\frac{4m\left(k{u}_0{}^2+m{v}_0{}^2+\gamma v_0{u}_0\right)}{4km-{\gamma }^2}}$ by fixing all the variable except $\gamma$.