How do I calculate the current in this circuit using Laplace transforms, with a time shift?
Not an answer! (This should be a comment but I don't have enough points). And the first part of the comment is a question:

Are you to assume that v1 has been active for a long time, so that the voltages on the inductor and capacitor have reached their steady state sinusoidal values? Or are you to assume that all voltages and currents were zero for all values of negative time?

If the second case, you must include both transient and steady state components to determine the cap voltage at t=.01. Too much work for me. If you are to assume the steady state sinusoidal situation, I give some rough calculations (using phasors, not Laplace, sorry)



zC = 1/jωC = 1/(j*2*π*100*.001) = -j*1.59 Ω

zL = jωL = j*2*π*100*.1 = j*62.8 Ω

zLC = 1/(jωC + 1/jωL) = -j*1.57 Ω

zRLC = 10.12 Ω /_-8.9° (-0.156 radians)

iR = (14.14V/10.12Ω) cos(100t + .156)

vC = v1 - vR

for t = .01, iR = +0.563A, so vR = 5.63V

vC = 7.64V - 5.63V = 2.01V



So I'm saying that (if we assume v1 has been steady state for a long time) the voltage on the cap just before the switch changes would be 2.01V. I didn't calculate it for the other case but I would not expect the full 7.64V across the cap. You might take another look at the value you calculated to be infinite.



[edit] Good catch and apologies Mr. Pina. Revision below. So the LC is resonant at 100 rad/sec. What a strange analysis problem.



zC = 1/jωC = 1/(j*100*.001) = -j*10 Ω  
zL = jωL = j*100*.1 = j*10 Ω  
zLC = 1/(jωC + 1/jωL) = 1/0 Ω  
zRLC = infinite  
iR = zero  
vC = v1 - vR

for t = .01, iR = 0A, so vR = 0V  
vC = 7.64V - 0 = 7.64V




I cannot review your Laplace analysis but I believe that (for first case above) the textbook solution is correct. The instantaneous current just after the switch is thrown should be

iC = (15.28V - 7.64V)/15Ω = 0.509A


If I understand, t' = 0 just after the switch is thrown? For t' = 0 that reduces to

0.509A = 1.56A*cos(91°) + transient(t'=0)  
0.509A = -0.027A + transient(t'=0)