The switch in Figure P9-43 has been closed for a long time and is opened at \(t=0\).

(a) Find the circuit differential equation in \(v_{\mathrm{C}}(t)\) and the initial conditions \(i_{\mathrm{L}}\) (o) and \(v_{\mathrm{C}}(\mathrm{o})\).

(b) The circuit parameters are \(L=10 \mathrm{H}, C=0.25 \mu \mathrm{F}, R_{1}=\) \(3 \mathrm{k} \Omega, R_{2}=10 \mathrm{k} \Omega\), and \(v_{\mathrm{S}}=12 u(t) \mathrm{V}\). Use Laplace transforms and MATLAB to solve for \(v_{\mathrm{C}}(t)\) for \(t \geq 0\).