a. from your clarificatiion, it sounds like you really have a linear, time varying system. So you don't have a transfer function representation, or any linear time invariant (LTI) representation.
b. I'm still confused by your Y(t). It kind of looks like the output of an LTI system, unless the alpha and wd are really functions of t, but just not shown as such. Also, Y(t) seems to be missing the effect of the input vSource. As you say, X1 and X2 are determined by initial conditions on y(t) and y'(t), but the system input should also have an effect on the output as well.
c. From what I gather, you have a second order, linear, time varying differential equation of the form:
z''(t)*Capacitance(t)*Inductance(t) + z'(t)*Inductance(t) = vSource(t) (where vSource(t) is a constant).
z''(t) = vSource/(C(t)*I(t)) - z'(t)/C(t)
You can implement this equation directly in Simulink using two integrators in series, one for z and the the other for z'. Set the initial condition on each as you desire. If you must use a time varying state space formulation, you can proceed as follows:
x = [z; z'].
A(t) =[0 1;0 -1/C(t)]
B(t) = [0; 1/(C(t)*I(t))]
C(t) = [1 0];
D(t) = 0;
Then set the initial conditions in your integrator block for x as [z(0) z'(0)]