uhh. what is PD? :(
Let me just clarify exactly the method I have learned and you can point at which step I use boundary conditions:
1) Say we have an RCL with I(0)=0, Q(0)=a driven by a time varying potential U(t) = εcos(ωt)
2) We write (where I is now complex I = I0exp(-iωt) and β is a phase)
εexp(-iωt) = RI0exp(-iωt) - iωLβI0exp(-iωt) - 1/(-iωC) I0exp(-iωt)
3) You can cancel out the exponential function and find the phase which can be found entirely in terms of the physical quantities L,C,R. The same goes for I0, the amplitude of the current.
4) You can now put it all together and take the real part of the current. It will be a sinusoidal function with determined amplitude and phase - which is one specific solution and is not a general solution.
WHERE did I apply the boundary conditions?