How does someone know if a differential equation has a special function solution?

Question

Is there some sort of list or table (like there is a table of integrals) that I could check my differential equation against whenever I encounter one?

Answer 1

What do you mean by a "special function"? There exist some kinds of differential equations that have solutions that can be written in terms of the "elementary" functions, polynomials, exponential, trig functions. There are far more kinds of differential equations that cannot be solved in terms those functions. We define a new function as a solution to such a differential equation and call it a "special function"- examples are "Bessel Functions", "Legendre functions", "Laguerre Polynomials" and others. There are, in fact, text books and lists of those but there can't be a list of "all" special functions because new "special functions" are being defined all the time.