The auxiliary equation for the given differential equation has complex roots. Find a general solution.
The auxiliary equation for the given differential equation has complex roots and its general solution is .
If the auxiliary equation has complex conjugate roots , then the general solution is given as:
Assume that is a solution to the given equation.
Since the given equation is of order two, differentiate role="math" localid="1654061985491" with respect to role="math" localid="1654061981089" twice:
Substitute and in the given equation to obtain:
Then the auxiliary equation is
Solve for the roots of the auxiliary equation
Since it has a complex conjugate of the form for and
Thus, the general solution is .
Series Circuit. In the study of an electrical circuit consisting of a resistor, capacitor, inductor, and an electromotive force (see Figure), we are led to an initial value problem of the form
where is the inductance in henrys, is the resistance in ohms, is the capacitance in farads, is the electromotive force in volts, is the charge in coulombs on the capacitor at the time , and role="math" localid="1654852406088" is the current in amperes. Find the current at time t if the charge on the capacitor is initially zero, the initial current is zero, role="math" localid="1654852401965" , and role="math" localid="1654852397693" .
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