How to find a solution to a homogeneous ordinary differential equation with non-constant coefficients (the coefficients are linear functions)?

Question

I am trying to find a solution to a differential equation in the form $$y'' + (ax+b)y' + (cx+d)y = 0,$$ where $a,b,c,d$ are constants. I only know how to solve when the coefficients are constants, and for the case of non-constant coefficients, I only know to solve when one of the solution is given. Is there any method for this kind of ODEs?

Answer 1

If you substitute $y(x)=e^{-\frac{1}{4}ax^2-\frac{1}{2}bx}\,u(x)$ in the ODE, you will obtain the following differential equation for $u$: $$ u''+(Ax^2+Bx+C)u=0, \tag{1} $$ where $A$, $B$ and $C$ are functions of $a$, $b$, $c$ and $d$. The solutions to $(1)$ are special functions known as parabolic cylinder functions.