Q$4-(25$ pts$)$ A capacitor with a capacitance $\backslash ($ C $\backslash )$ is fully charged by a battery with an emf value of $\backslash ($ V $\backslash ).$ Then, the polarity of the battery is reversed to reach the configuration shown in the figure, and the switch S is closed at time $\backslash ($ t$=0\backslash ).$ Express your answers to the questions below in terms of the variables $\backslash ($ V$,$ C$,$ R$,$ t $\backslash ).$

a$)(4$ pts$.)$ What are the initial and final charges on the capacitor's upper plate?

b$)(10$ pts$.)$ Kirchhoff's loop law can be expressed as a differential equation for the charge $\backslash ($ q$($t$)\backslash )$ on the upper plate as: $\backslash (\backslash$frac$\{$d q$\}\{$d t$\}+$A q$+$B$=0\backslash ).$ Find $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash ).($No partial grade for answers with the wrong sign.$)$

c$)(11$ pts$.)$ Find an expression for $\backslash ($ q$($t$)\backslash ).$