It's been too long guys. I've given this ODE lots of thought and still no cigar. 1. The problem statement, all variables and given/known data We are given the following ODE: $$ (x-a)y''-xy'+a^2y = a(x-1)^2e^x $$ and knowing that y=e^x is a solution to the homogenous equation, find the possible values of a. Next part: Using the obtained values, calculate the solution (y(x)) so that it's limit as x→ -∞ is contained within a finite set. (Also, let the solution be continuous in said set) 2. Relevant equations 3. The attempt at a solution First I replaced the known base of the homogenous solution in the equation. I got that a can be either 0 or 1. From here I investigate for the case if a=1 Knowing a base of the homogenous solution, I set out to find the missing one knowing a ODE of order 2 has 2 bases for the Homogenous solution. The way I tried to find it was writing it in the form of y2=α(x)ex deriving it 2 times, then replacing each expression in it's respective place in the homogenous ODE. Using the principle of superposition I rearranged the resulting expression as follows: $$ e^x \left( x(\alpha ''+\alpha ' )-( \alpha '' +2\alpha ') \right)=0 $$ Where α is an unknown function. From here we get two seemingly contradictory equations, one points to α being e-x and the other to e-2x, each giving me a different base of the homogenous equation. This cannot be, I must be doing something wrong. But what?