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Differential Equations : Solutions, Electric Circuit and Step Function

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Find the general solution of each of the differential equation y ''- y ' = x2 . If the solution is not valid over the entire real axis, describe an interval over which it is valid.

If k is a nonzero constant, prove that the equation y ''+ k2y = R(x) has a particular solution
1 y given by
.....
Find the general solution of the equation y ''+ 9y = sin3x .

The current I(t ) at time t flowing in an electric circuit obeys the differential equationI ''(t ) + I(t ) = G(t ) , where G is a step function given by G(t ) = 1 if 0 ≥ t ≥ 2p , G(t ) = 0 for all other t.
Determine the solution which satisfies the initial conditions I(0) = 0,I '(0) = 1

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Solutions, electric circuit and step functions are investigated and discussed in the solution. The solution is detailed and well presented.

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