(a) In the circuit shown in Figure P2.61 (a), x(t) is the input voltage. The voltage y (...

(a) In the circuit shown in Figure P2.61 (a), x(t) is the input voltage. The voltage y (t) across the capacitor is considered to be the system output.

(i) Determine the differential equation relating x(t) and y(t).

(ii) Show that the homogeneous solution of the differential equation from part (i) has the form Specify the values of to and cot.

(iii) Show that, since the voltage and current are restricted to be real, the natural response of the system is sinusoidal.

(b) In the circuit shown in Figure P2.61(b), x(t) is the input voltage. The voltage y(t) across the capacitor is considered to be the system output.

(i) Determine the differential equation relating x(t) and y(t).

(ii) Show that the natural response of this system has the form , and specify the value of a.

(c) In the circuit shown in Figure P2.61(c), x(t) is the input voltage. The voltage. y(t) across the capacitor is considered to be the system output.

(i) Determine the differential equation relating x(t) and y(t).

(ii) Show that the homogeneous solution of the differential equation from part (i) has the form and specify the value of a.

(iii) Show that, since the voltage and current are restricted to be real, the natural response of the system is a decaying sinusoid.