My task is to solve circuit below using Laplace transform:

Firstly, I solved it using complex numbers and I got:

I'm sure it's correct. Here is my solution using Laplace transform:

It's obvious that solutions I got using complex numbers and Laplace transform are different. Where did I make mistake?

 

My first guess would be that you aren't using the same driving function in both cases. Hint: The transform you used is the single-sided Laplace transform.

 

Should I use two-sided Laplace transform? Just to mention that I don't have any problem in RC circuit analysis using single-sided Laplace transform.

 

But a single-sided transform means that the function being transfomed is taken to be identically zero for all time t<0. But the steady state analysis you did first assumes that the sinusoidal voltage is, always has been, and always will be sinusoidal.

Do you expect a circuit that has been running in sinusoidal steady state to have the same response for t>0 as the same circuit that has a sinusoidal source suddenly turned on at t=0?

 

Does it mean that my calculations include both transient and steady-state response?

 

I would expect it to.

But you answer has more than one problem. Notice that it is always positive. Does that make sense?