Goal is to solve for the voltage u₂(t)

I'm somewhat confused by this. I solved the transfer function Z(s) earlier in the top circuit, which is simply just Z. The voltage across Z in the top circuit is $$u_1(t)=5e^{(-100t)}u(t)$$ and the current source is j(t) = u(t), where u(t) is the step function. Z is a circuit system itself with no independent sources and all initial values are zero.

Using basic principles for u₂(t) you obtain e(t)·Z/(R+Z), where again e(t) = u(t), so it's u(t)·Z/(R+Z).

Obviously, just from that equation you can't solve for voltage. I'm thinking this has something to do with the frequency response, but I don't know how I'm supposed to use it for this problem.

Z(s) is the same in both circuits.