P$7-$ The switch in the circuit below is closed at $t=0,$ and the capacitor is initially uncharged.

a$)$ Using Laplace transform methods, determine the current $i(t)$ and the voltage $v_C(t)$ for $t>0$;

b$)$ Evaluate $i(0)$ and $v_C(0)$ and compare these values with those obtained from the circuit analysis using the circuit model at $t=0+.$

c$)$ Repeat the analysis above if the capacitor is initially charged to a value of $10$ V $.$

P$8-$ The switch in the circuit below is closed at $t=0,$ and the circuit is initially relaxed. Use Laplace methods to calculate the current $i(t)$

No AI answers or looking answer handwritten is prefered