I am given the following: xy'' - (x+1)y' + y = 0. I need to find the solution.

Even a casual glance shows that at least one solution is y = e^x: if y''=y'=y, the equation reduces to an identity. The problem is that when I go through the steps, I keep getting the following recursion equation:

k^2(a sub(k-1)) - k(a sub(k)) = 0 at each level of k. That's clearly wrong; it causes the final solution to have factorials in the numerator instead of the denominator where they belong. But I have no idea what I am doing wrong. I have checked and rechecked the steps and keep coming up with the same thing.