
Power Series Solution
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!

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.