What non-algebraic methods do we have to solve this equation?

Question

I'm recently studying homogeneous linear second order ODEs, and got into the integrating factors technique. For every equation, if the integrating factor is known, we can reduce its order provided that the equation contains consecutive derivatives, such as these: $$A(x)\frac{d^{n+1}}{dx^{n+1}}y+B(x)\frac{d^n}{dx^n}y=0$$
Which can be reduced to $$A(x)\frac{d}{dx}p(y)+B(x)p(y)=0$$
Which can be solved by the integrating factor method, with $p(y)=\frac{d^n}{dx^n}y$, by repeatedly integrating . But i've noticed some impossibilities in applying a similar process to non-consecutive derivatives of a function... which appears to imply that the simplest kind of ODE equation that we cannot solve differ-algebraically is the second order ODE: $$y''+A(x)y=0$$ Is this true? If it is, what other methods can we apply, other than the differ-algebraic solution?

Answer 1

One way to solve it is with series. If $A(x)$ is a rational function, there will be a linear recurrence for the coefficients of the Maclaurin series of a solution. In some cases that can be solved, and in some cases it can lead to expressing the solution using special functions.

Another way is to go through a list of special functions and see which satisfy this one (or a second-order differential equation that can be made equivalent to this one).