At t = 0 t = 0 , the switch in Figure P6-51 is closed and thereafter the voltage across the capacitor is v C

At $t=0$, the switch in Figure P6-51 is closed and thereafter the voltage across the capacitor is
$$v_C\left(t\right)=(10+10,000t)e^{-8000t}V$$
Use MATLAB to solve all of the following problems.
(a) Use the capacitor's $i-v$ characteristic to find the current $i\left(t\right)$ for $t\ge 0$.
(b) Use the inductor's $i-v$ characteristic and $i\left(t\right)$ to find $v$ $L\left(t\right)$ for $t\ge 0$.
(c) Use $v_C\left(t\right),v_L\left(t\right)$, and KVL to find the voltage $v\left(t\right)$ delivered to the rest of the circuit.
(d) The $v\left(t\right)$ found in (c) should be proportional to the $i$ ( $t$ ) found in (a). If so, what is the equivalent resistance looking into the rest of the circuit?
(e) On the same axes, plot $v_C\left(t\right),v_L\left(t\right)$, and $v\left(t\right)$. Use a different color for each waveform. Use the plots to verify KVL for the circuit.

25 F t=0 2.5 mH i(t) Xo voo - + VL (1)- + vc(1) v(t) VI Rest of the circuit