How to solve the following differential equation $$y''(x)+Q(x)y(x)=0$$
And how to find exact solution $y(x)$ in terms of special functions?
2026-04-08 05:48:50.1775627330
Solving differential equation $y''(x)+Q(x)y(x)=0$
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Your question is not precise enough formulated... Are you looking at ANY $Q(x)$, or you are interested in polynomial $Q(x)$, etc...
Such type of equation appears routinely in quantum mechanics, when calculating eigenstates and eigen-energies of one-dimensional systems, or corresponds to the three-dimensional systems via the so-called radial Schroedinger equation (spherical coordinates, or alike)...
There are particular cases when this is solvable in closed form (special functions, etc), but for a general solution, only approximations are available (see the WKB method and similar).
The literature is huge on this topic (regarding analytical solutions): the WKB, operational calculus (again, hard to tackle the general case), Laplace transform technique, homotopy method... You can search on the web for them, and you find plenty of papers...
If you look at a numerical solution, again, depending on the characteristics of $Q(x)$, many numerical issues can appear the hinder obtaining the solution to, say, machine precision in reasonable computer time and memory.
All in all, you need be more specific about $Q(x)$, otherwise the topic is too large to be addressed in a simpler manner...