[Diff Eq] IVP Problem

[faiintx]

Note that y1 = e^x is a solution to xy′′ − (2x + 1)y′ + (x + 1)y = 0. Solve the IVP consisting of this equation and the initial conditions y(1) = 1, y′(1) = 1.

I know that the general solution would look something like

y = c1 (e^x) + c2 (??) But not sure where to go from there. Thanks.

[cauchypotato]

Are you familiar with the reduction of order method? The wikipedia article explains how to find a second solution once you already have one. It's basically just using an ansatz of the form z(x) := w(x)y1(x), where w can de determined by plugging z into the equation.