Chapter 30, Problem 42E

(a)

To determine

To show: Whether equation in $LCR$ circuit, $q=A{e}^{-\left(R/2L\right)t}\cos \left({\omega }^{\prime }t+\varphi \right)$ is the solution of the equation $\frac{d^{2}q}{d{t}^2}+\frac{R}{L}\frac{dq}{dt}+\frac{q}{LC}=0$.

(b)

To determine

To show: The constants $\tan \varphi =-\frac{R}{2L\sqrt{\left(1/LC\right)-\left(R^2/4L^2\right)}}$ and $A=\frac{Q}{\cos \varphi }$ from the equation $q=A{e}^{-\left(R/2L\right)t}\cos \left({\omega }^{\text{'}}t+\varphi \right)$.