$[$DC RLC Circuit$]$ Consider the circuit below, with a resistor, capacitor, and inductor. At time

$=0,$ switch $$ is open, the currents are all zero, and there is no charge on the capacitor. We will call $$

the current through the resistor, $1$ the current through the inductor, $2$ the current through the

capacitor, and $2$ the charge on the capacitor, all of which are functions of time.

$($a$)$ Immediately after the switch is closed, what are $,1,2,$ and $2?$

$($b$)$ After a very long time, what are $,1,2,$ and $2?$

$($c$)$ After a very long time, the switch is opened again. Immediately after the switch is opened again,

what are $,1,2,$ and $2?$

$($d$)$ Use Kirchoff$$s laws to find a differential equation for $2(),$ the charge on the capacitor after the

switch is re$-$opened.

Assume $=0$ is the time when the switch is opened. The charge on the capacitor takes the form:

$2()=$ sin $+$ cos $$

$($e$)$ What is the value of $$ that satifies the differential equation from part $($d$)?$

$($f$)$ Find the constants $$ and $$ in terms of the given circuit parameters, $,,,$ and $.$

Tip: What are your $$boundary conditions$?$ I.e$.,$ what do you know about the charge on the

capacitor, etc?