How to Find DC Step Response of Semi-Parallel RLC Circuit
one more for you:

\$i_{cR_2} = C \frac{dV_c}{dt} = \frac{V_a-V_c}{R_2} = f(V_a,V_c) ----(3) \$ 

Now you have three independent equations and three unknowns (\$i_L,V_c,V_a\$) to solve everything.