Chapter 10, Problem 9P

A sinusoidal voltage $v_s\left(t\right)=10\sin \left(0.01t\right)\text{V}$ is applied to the RC circuit shown in Fig. P10.6. The voltage $v\left(t\right)$ across the capacitor satisfies the first order differential equation

$RC\frac{dv\left(t\right)}{dt}+v\left(t\right)=10\sin \left(0.01t\right)$.

(a) Find the transient solution, $v_{tran}\left(t\right)$. What is the time constant of the response?

(b) Find the steady-state solution $v_{ss}\left(t\right)$ and plot one cycle of the response. Note: One of the two terms in the steady-state solution is small enough to be neglected.

(c) If the initial voltage across the capacitor is $v\left(0\right)=0$, determine the total solution $v\left(t\right)$.