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Capacitor Discharge

An application of homogeneous differential equations

A first order homogeneous differential equation

has a solution of the form :
.

For the process of discharging a capacitor C, which is initially charged to the voltage of a battery V_b, the equation is

Using the boundary condition and identifying the terms corresponding to the general solution, the solutions for the charge on the capacitor and the current are:

.

Since the voltage on the capacitor during the discharge is strictly determined by the charge on the capacitor, it follows the same pattern.

.