a question about circuits and diferential equation

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Consider the DC circuit of Figure $ 4$. Inductance L satisfies $L = (R^2) C / 2$. Calculate:

a) The differential equation for the charge $ Q (t) $ contained in the capacitor;

b) The solution of diferential equation of the preceding item;

c) the power dissipated in the resistance and the energy stored in the capacitor for sufficiently long times so that only the stationary solution is present.

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I am going to give you a sketch on how you can find the solution.

As I stated in the comments, notice that this is not an RLC circuit, because C and L are in parallel whereas they are in series in an RLC circuit.

Since C and L are in parallel, we have

$$v_C(t) = v_L(t)$$

and

$$i(t) = i_R(t) = i_L(t) + i_C(t)$$

From Kirchhoff's voltage law:

$$v_R(t)+v_L(t) = v_R(t)+v_C(t) = -\epsilon$$

Moreover,

$$v_L(t) = L \frac{di_L(t)}{dt}$$

and

$$ v_C(t) = \frac {Q(t)}{C} = \frac 1 C \int_{-\infty}^{t} i_C(\tau) d \tau$$

This is all you need to solve the problem.