exponential term in RLC sinusoidal analysis

Question

I was learning to work with RLC circuits with sinusoidal excitations and the book I was dealing with showed me that the responses of these passive elements to sinusoidal excitation were themselves purely sinusoidal and hence phasor digarams were the best way to deal with these elements, but then I tried solving a simple series RLC circuit using laplace/complex frequency analysis and I landed myself a sine term, a cosine term and a real exponential term. I don't know what or why I am getting that real exponential term, I tried to do it over and over and I got the same results. I am pretty sure my calculations are correct but the book mentions nothing about the exponential term. Kindly pitch in...

This is the circuit I was talking about

^{simulate this circuit – Schematic created using CircuitLab}

Here is an image for a KVL and laplace transformation

Here is an image for the inverse after separation. I dont think there is a mistake

Answer 1

That real exponential term represents the transient response of the system. Generally, when doing steady-state sinusoidal analysis, you can simply ignore the transient response altogether, since the real part of the exponent is usually a negative value times t (time), which goes to zero as t → ∞. If not, it means the system is unstable to begin with.