Chapter 4.4, Problem 26P

Consider the initial value problem

$\begin{array}{ccc}my\text{'}\text{'}+\gamma y\text{'}+ky=0,& y\left(0\right)=y_0,& y\text{'}\left(0\right)=v_0\end{array}$.

Assume that ${\gamma }^2<4km$.

Solve the initial value problem.

Write the solution in the form $y\left(t\right)=R\exp \left(-\gamma t/2m\right)\cos \left(vt-\delta \right)$. Determine $R$ in terms of $m,\gamma ,k,y_0$ and $v_0$.

Investing the dependence of $R$ on the damping coefficient $\gamma$ for fixed values of the other parameters.