Once t > 0, the current source does not deliver any current anymore.
We have a series RLC circuit. We can notice that we have
Thus we know that we will have s1 = s2 = -α = - R/(2*L) = 2000
And we also know that the answer will be a critically damped response,
thus
Vc(t) = (A1 + A2t)*exp( - 2000*t)
We are looking for A1 and A2.
For t = 0 we have Vc(0) = A1, and by the equivalent source we have a
voltage of
V = R*I = 100*30.10-3 = 3 V, so A1 = 3 V
For A2, we need to use the derivative. We have
dV(t)/dt = (3 + A2t)*( - 2000*exp( - 2000*t))
+ A2*exp( - 2000*t) = i(t)/C
For t = 0 we have 3*( - 2000) + A2= i(0)/C, but i(0) = 0 A
Finally we get Vc(t) = (3 + 6000*t)*exp( - 2000*t)