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# Find the resonance frequencies of a circuit governed by a given differential equation

The current in an electric circuit obeys the differential equation

for positive real numbers and . The solutions to this equation can be expressed as

where and are constants which depend on and and

If there exists a value of which maximizes then the value is called a resonant frequency of the circuit.

1. Find all resonant frequencies when .
2. Find all values of such that the frequency has a resonant frequency.

1. The homogeneous equation is of the form where and . This gives us . Therefore, . Hence, the general solution of the homogeneous equation is

Now, to find a particular solution of we let

Plugging these into the differential equation we have

Evaluating at and we obtain the two equation

Solving for and we obtain

Thus, we have the particular solution

Therefore, the general solution is

Where,

This implies

Furthermore, from a previous exercise (Section 2.8, Exercise #9) we know for we have

where

So, to maximize we minimize . Setting the derivative equal to 0 we have

Hence, the resonance frequency is

2. In order to have a resonance frequency, we must have a value of such that is maximized. Hence,