General solution of second order linear differential equation

Question

Is it possible to find a general solution of: $$y''+p(x)y'+q(x)y=0$$

where $p$ and $q$ are functions of $x$. Or is it impossible to solve just like the general quintic equation?

Answer 1

Maybe surprisingly, the answer is no - even for really simple $p$ and $q$.

For example, $y''-xy=0$ (i.e. $p(x)=0$ and $q(x)=-x$) is called the Airy equation. It has a solution that not only fails to be an elementary function, but in fact it cannot even be obtained from a finite number of algebraic operations and anti-differentiations from an elementary function (unlike, for example, the error function).

https://en.wikipedia.org/wiki/Airy_function

The general study of whether these (and more general) differential equations can be solved is called differential Galois theory.

Here's links to handouts from a class I taught a few years ago:

https://mathsci2.appstate.edu/~cookwj/courses/math4010-spring2016/math4010-spring2016-differential_algebra.pdf

https://mathsci2.appstate.edu/~cookwj/courses/math4010-spring2016/math4010-spring2016-differential_galois_theory.pdf