20 sin 30t Suggesting a solution of the form i p (t) = A sin(30t + ) The derivative of this expression is di p = 30A cos(30t + ) dt Now use the following trig identities sin(x + y) = sin x cos y + cos x sin y cos(x + y) = cos x cos y sin x sin y To write i p (t) = A sin(30t + ) = A sin 30t cos + A cos 30t sin di p = 30A cos(30t + ) = 30A cos 30t cos 30A sin 30t sin dt Then we have di p + 3i p (t) = 30A cos 30t cos 30A sin 30t sin dt + 3A sin 30t cos + 3A cos 30t sin Grouping terms and setting equal to the inhomogeneous term due to the voltage source we have A cos 30t(30 cos + 3 sin ) + A sin 30t(3 cos 30 sin ) = 5 sin 30t Since there is no term involving cos 30t on the right-hand side, it must be true that 30 cos + 3 sin = 0
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