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