PX157 - E4d - solving as an ODE

\begin{matrix} V_{i n} (t) = R I + L \frac{d I}{d t} + \frac{Q}{C} \\ L \frac{d^{2} I}{d t} + R \frac{d I}{d t} + \frac{1}{C} I = \frac{d V_{i n}}{d t} \end{matrix}

\begin{aligned} I (t) & \sim e^{λ t} \\ L λ^{2} + R λ + \frac{1}{C} & = 0 \\ λ_{\frac{1}{2}} & = - \frac{R}{2 L} \pm \frac{1}{2 L} \sqrt{R^{2} - \frac{4 L}{C}} \end{aligned}