For Prob. 8.51.

Step: 1

Refer to Figure 8.98 .
Initially at, the switch is closed for a long time. The circuit has reached to steady state and the inductor acts like a short circuit, while the capacitor acts like an open circuit.
The equivalent circuit diagram is shown in Figure 1.

Step: 2

In Figure 1, there is a short circuit across the resistor, . Thus, no current flows through the resistor, . Redraw the modified circuit diagram.

Step: 3

From Figure 2, the initial voltage across capacitor is,

The capacitor voltage does not change instantaneously.

From Figure 2, the initial current through the inductor is,

The inductor current does not change instantaneously.

Step: 4

For , the switch is opened. The equivalent circuit diagram is shown in Figure 3.

Step: 5

From Figure 3, it is a source-free parallel LC circuit.
Write the formula for neper frequency:

In Figure 3, there is no resistance in parallel with the energy storing elements.
Therefore, the value of neper frequency is,

Write the formula for resonance frequency:

From the above calculations, .
Therefore, it is an under damped system.
 

Step: 6

For an under damped system, the expression for voltage is,
 

Here,

Substitute  for in the equation.

Therefore, the expression for voltage across the capacitor is,

Substitute  for in the equation.

…… (1)

Step: 7

Determine the value of.
Substitute for  in equation (1).

Substitute  for  in the equation.

From Figure 3, the current flowing through the capacitor is,

Step: 8

Differentiate the voltage expression with respect t, to determine the value of.

Substitute  for  in the equation.

Substitute  for  in the equation.

Step: 9

Recall the equation (1).

Substitute  for and  for  in the equation, to derive the expression for voltage across the capacitor.

Thus, the expression for voltage across the capacitor is,
.