Determining energy stored in capacitor and inductor in RLC circuit

Question

I've been stuck on the following homework problem for a few hours now with little progress.

As you can see, it's a relatively simple RLC circuit with a couple independent sources and a voltage-controlled voltage source. Despite this, I have been unable to solve for mesh currents and nodal voltages despite repeated attempts at tackling the problem.

From what I understand, the energy stored by a capacitor over $[t_0,t]$ follows from $i=C\frac{dV}{dt}$ since $W=\int^t_{t_0} Pdt=\int iVdt = \int_{t_0}^t CV\frac{dV}{dt}dt=C\int_{V_0}^V V\ dV=\frac12C(V^2-V_0^2)$.

Similarly, for an inductor, $V=L\frac{di}{dt}$ so $W=\int_{t_0}^tLi\frac{di}{dt}dt=L\int_{i_0}^i i\ di=\frac12L(i^2-i_0^2)$.

Unfortunately, I haven't been able to find the above $V(t),i(t)$. $V_0,i_0$ were however trivial to solve for using a DC-equivalent circuit.

How does one go about solving for these functions?

Answer 1

A couple of suggestions:

(1) the EE stackexchange site a better home for this question

(2) simply solve for the voltage across the capacitor and the current through the inductor. Once you have those, the energies stored, as a function of time are just

$$W_L(t) = \frac{L}{2}i^2_L$$

and

$$W_C(t) = \frac{C}{2}v^2_C$$

Since this is evidently a DC circuit in steady state (big hint here), the voltages and currents are constant so the stored energies are constant.