Problem $4.$ The initial voltages on capacitors $C_1$ and $C_2$ in the circuit shown below have been established by sources not shown. The switch is closed at $t=0.$

$($a$)$ Find $v_1(t),v_2(t),$ and $v(t)$ for $t0$ and $i(t)$ for $t{0}^{+}.$

$($b$)$ Calculate the initial energy stored in the capacitors $C_1$ and $C_2.$

$($c$)$ Determine how much energy is stored in the capacitors as $t.$

$($d$)$ Show that the total energy delivered to the $250k$ resistor is the difference between the results obtained in $($b$)$ and $($c$).$

Fig. $4.$