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 $${V_C}\left( t \right)$$

Viewed from the terminals $$A, B$$ the following circuit shown in Figure can be reduced to an equivalent circuit of a single voltage source in series with a single resistor with the following parameters:

The effective inductance of the circuit across the terminals $$A,B$$ in the Figure shown below is

In the circuit shown in Figure, it is desired to have a constant direct current $$i(t)$$ through the ideal inductor $$L.$$ The nature of the voltage source $$v(t)$$ must be