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.
- Find all resonant frequencies when .
- Find all values of such that the frequency has a resonant frequency.
- 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
Furthermore, from a previous exercise (Section 2.8, Exercise #9) we know for we have
So, to maximize we minimize . Setting the derivative equal to 0 we have
Hence, the resonance frequency is
- In order to have a resonance frequency, we must have a value of such that is maximized. Hence,