laplace transform is very useful for finding various parameters and solutions to complex circuits and equations. u can get some idea abt it from here. http://en.wikipedia.org/wiki/Laplace_Transform u may for now have to solve the above equation using solution methods for differential equations.
I think this is easiest with phasors... 4*I + (8*I)/(j*w) - 3*j*w*I = 50<75 w = omega = 2 4*I - (j*8*I)/2 - j*6*I = 50<75 add the imaginary terms 4*I - j*65*I = 50<75 convert to polar form 65.123<-86.5 * I = 50<75 divide the magnitudes and subtract the angles I = 0.7678<161.5 i = 0.7678*cos(2t + 161.5 degrees) A This is very similar to the Laplace transform...basically "s" becomes "jw", like Fourier. But w is constant, so it is in "phase-domain" instead of frequency-domain. You might want to double-check my algebra/arithmetic, I'm not really that awake.