Pi Circuit

Circuit equations using Kirchoff's laws:

$I_S = I_1 + I_L$ $I_L = I_2 + I_R$ $\dot{q}_1 = I_1$ $\dot{q}_2 = I_2$ $-V_S + I_SR_S + \frac{q_1}{C_1}=0$ $-\frac{q_1}{C_1} + L\dot{I}_L + \frac{q_2}{C_2}=0$ $I_RR = \frac{q_2}{C_2}$

This time, I am omitting the details of recasting equations into a state-space form. Initial conditions:

$q_1(0) = q_2(0) = I_L(0) = I_2(0) = I_R(0) = 0 \ ; \ I_1(0) = 10$

Numerically integration gives the required answer as $\boxed{7.777}$ . The value $\frac{10}{3}$ happens to be the steady-state source current which can be found by simply inspecting the circuit.

I will update the solution if more details are requested. As for the plot of source current vs. time, I have not attached one, but it is easy to conclude by inspection that the current $I_S$ starts at a value of $10A$ at $t=0$ and decays to a steady-state value of $\frac{10}{3}A$ after a long time. Due to the dynamics of the circuit, it is observed that the decay of current happens in an under-damped manner.