# Applications of Linear Differential Equations to Electric Circuits

• ### Prerequisites :

Differential Equations of First Order and First Degree

Linear Differential Equations of Higher Order

There are 3 basic components of an electric circuit, where a change in voltage is possible. They are:

1) Resistance (${R}$), voltage drop = ${i R}$, we saw this in Ohm’s law.

2) Capacitance (${C}$), voltage drop = ${\frac {1}{C} \times \int i dt}$

3) Inductance (${L}$), voltage drop = ${L \times \frac {di}{dt}}$

Note that the electric current ${i}$ is the rate of flow of charge ${q}$, hence, ${i = \frac {dq}{dt}}$.

To solve a differential equation, we use the Kirchoff’s voltage law, which states that the sum of all the voltages around a loop is equal to zero.

The general circuit will consist of all 3 elements, ${R,L}$ and ${C}$ as well as the voltage source, ${E}$. The D.E. will be

${L \frac {di}{dt} + i R + \frac {1}{C} \int i dt = E sin (\omega_a t)}$

Expressing in terms of the amount of charge, ${q}$,

${L \frac {d^2q}{dt^2} + R \frac {dq}{dt} + \frac {q}{C} = E sin (\omega_a t)}$

${\omega_a}$ is the frequency of applied voltage.

Resonance is a special condition in ${R-L-C}$ circuits, when the imaginary parts of impedances due to inductor and capacitor cancel each other. It occurs when the applied frequency ${\omega_a}$ becomes equal to the natural frequency, given by

${\omega_n = \frac {1}{\sqrt {LC}}}$