Yes you should express in terms of its power series. It shouldn't be too hard to combine all the series together and so, the question is no different from a homogenous equation.
We are given the IVP y'' - xy = e^x , y(0)=1, y'(0)=0 and we are asked to find a recurrence relation and the first four nonzero terms in the power series solution of this IVP. The problem is that all the problems of this type that i have encountered are the homogeneous equations. Here since the RHS is not zero, should we write the Taylor series expansion of e^x and then equate the coefficients after finding the a_2? I did so and found a_2 to be 1/2. Is it right? And what is the solution in case to check my answer? Thanks for your help!