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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, 