6th Order RLC

I will post my solution as well, since there are minor stylistic differences relative to the solution given by @Karan Chatrath . The state space representation is:

$V_{L1} = V_S - R_1 I_{L1} - V_{C3} = L_1 \dot{I}_{L1} \\ V_{L2} = V_{C3} + V_{C1} - R_3 I_{L2} = L_2 \dot{I}_{L2} \\ V_{L3} = V_{C3} = L_3 \dot{I}_{L3} \\ I_{R2} = \frac{V_S - (V_{C3} + V_{C1})}{R_2} \\ I_{C1} = I_{R2} - I_{L2} = C_1 \dot{V}_{C1} \\ I_{R4} = \frac{V_{C3} - V_{C2}}{R_4} \\ I_{C2} = I_{R4} = C_2 \dot{V}_{C2} \\ I_{C3} = I_{L1} + I_{C1} - I_{R4} - I_{L3} = C_3 \dot{V}_{C3}$

Isolate the time derivatives of state variables $(I_{L1}, I_{L2}, I_{L3}, V_{C1}, V_{C2}, V_{C3})$ and numerically integrate. Then the source current is:

$I_S = I_{L1} + I_{R2}$

Currents and charges on the capacitor are as shown in the diagram. As charge accumulates in the capacitors, one obtains:

$\dot{Q}_1 = I_1 \ ; \ \dot{Q}_2 = I_2 \ ; \ \dot{Q}_3= I_3$

From Kirchoff's current law:

$I = I_6 + I_7$ $I_7 = I_1 + I_5$ $I_1 + I_6 = I_2 + I_8$ $I_8 = I_3 + I_4$

From Kirchoff's voltage law, ignoring all passive element notation, as they all have a value of unity, one gets:

$-V_S + I_7 + I_5 + \dot{I}_5 = 0$ $\dot{I}_6 + I_6 = I_7 + Q_1$ $\dot{I}_4 = I_2 + Q_2$ $\dot{I}_4 = Q_3$ $Q_1 + I_2 + Q_2 = I_5 + \dot{I}_5$

This system of equations can be re-arranged to obtain a sixth-order state-space form. However, this re-arrangement has been avoided. These differential-algebraic equations are numerically integrated to obtain the required result which is as follows:

The required answer turns out to be:

$\boxed{Q \approx 20.4146}$