It's a second order linear ODE so it's going to have two linearly independent solutions, and the general solution will be a linear combination of these. Probably the easiest way to handle the recurrence relation is to get the two lin. ind. solutions as follows: for the first one, take a_0=1 and a_1=0. For the second one, take a_0=0 and a_1=1. That will have the effect of "separating the a_0 and a_1 terms", as you wanted. You will get two solutions, and you can then form a lin. comb. of these to get the general solution.