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In the process of finding transient response for circuits with AC excitation using differential equations, we use the method of complementary functions and particular solution, but I read earlier that the solution for total response (transient and steady state) of a circuit is the sum of the complementary function (which is the transient response) and the particular solution (the steady state response) of the corresponding differential equation. Then for the given circuit, shouldn't the particular solution be equal to zero since we're only considering the transient response? enter image description here

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For the given circuit, the particular solution cannot be zero because the circuit is being excited by a sinusoidal source. Its derivative is also sinusoidal. In contrast, the transient response is determined by the initial voltage, which is a constant and whose derivative is zero. It would be easier to determine the transient and steady-state responses of this circuit using Laplace transforms and partial function expansion.

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