The switch in Figure P9-41 has been open for a long time and is closed at \(t=0\). The circuit parameters are \(R=500\) \(\Omega, L=2.5 \mathrm{H}, C=25 \mu \mathrm{F}\), and \(V_{\mathrm{A}}=500 \mathrm{~V}\).

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

(b) Use Laplace transforms to solve for \(v_{\mathrm{C}}(t)\) for \(t \geq 0\).

(c) Simulate the circuit in Multisim to plot your resulting waveform.