Is 2nd-order linear ODE with quadratic coefficients solvable?

Question

Consider a general 2nd-order ODE $$\alpha_2 y''(x) + \alpha_1 y'(x) + \alpha_0 y(x)=0$$ where coefficients are all real quadratic polynomials of $x$ $$\alpha_n=\sum_{i=0}^{2}{a_i^{(n)} x^i}.$$ Is it possible to have analytic and especially closed-form solutions (including special functions)? If not, under what assumption of the coefficients (like which one should be zero or so) it is known to have?

Answer 1

$$\begin{cases} \alpha_2(x) y''(x) + \alpha_1(x) y'(x) + \alpha_0(x) y(x)=0\\ \alpha_n(x)=\sum_{i=0}^{2}{a_{i,n} x^i} \end{cases}$$ This linear second order ODE includes a big number of parameters ( Height independent ). As a consequence the sub-cases are numerous :

Airy, Bessel, modified Bessel, spherical Bessel, Chebyshev, Confluent hypergeometric, Gauss hypergeometric, Hermite, Jacobi, Laguerre, Legendre, Gegenbauer, Weber, Whittaker, … and many other.

A lot of special functions were defined allowing to express the solutions of those ODEs on closed form. But all sub-cases were not studied and the relevant special functions are not standardized.

One can understand that your question is much to wide and that no general answer covering all cases can be given.

If you are concerned by this kind of general problem, in a first step you should restrict you query to the ODEs with only 3 or 4 independent parameters instead of 8. This would be a big job but more realistic nowadays.