In the given circuit, the capacitor is initially charged to $$12$$ $$V. $$ Find the mathematical expression for the voltage across the capacitor $${{V_C}}$$ after closing the switch at $$t = 0.$$

The switch in the following circuit, shown in Fig. has been connected to the $$12V$$ source for a long time. At $$t=0,$$ the switch is thrown to $$24$$ $$V$$.

$$L = 2\,H,\,\,{R_1} = 10\,\Omega ,\,\,{R_2} = 2\,\Omega ,\,\,C = 0.25\,\mu F$$
$$(a)$$ Determine $${i_L}\left( 0 \right)$$ and $${V_C}\left( 0 \right)$$
$$(b)$$ Write the differential equation governing $${V_C}\left( t \right)$$ for $$t>0$$
$$(c)$$ Compute the steady state value of $${V_C}\left( t \right)$$

In the circuit shown in Fig., $${e_g}\left( t \right) = 2.5t$$ volts. What are the values of $$i(t)$$ and $${V_L}\left( t \right)$$ at $$t=4$$ seconds?