## Identifying a second-order ODE

Is there a general solution to

$$\frac{d}{dt}\left[p(t)\frac{dx(t)}{dt}\right] + q(t)x(t) = 0$$

for $x(t)$ when $p(t)$ and $q(t)$ are arbitrary functions? Better yet, does this question have a name, or some identifier, that I could look in to? It might appear more familiar written as

$$\left[p(t)x^\prime\right]^\prime + q(t)x = 0$$
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 Quote by Undoubtedly0 Is there a general solution to $$\frac{d}{dt}\left[p(t)\frac{dx(t)}{dt}\right] + q(t)x(t) = 0$$ for $x(t)$ when $p(t)$ and $q(t)$ are arbitrary functions?
No, there is no genertal solution
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 Quote by Undoubtedly0 Is there a general solution to $$\frac{d}{dt}\left[p(t)\frac{dx(t)}{dt}\right] + q(t)x(t) = 0$$ for $x(t)$ when $p(t)$ and $q(t)$ are arbitrary functions? Better yet, does this question have a name, or some identifier, that I could look in to? It might appear more familiar written as $$\left[p(t)x^\prime\right]^\prime + q(t)x = 0$$
Look up Sturm-Liouville problems or equations.

## Identifying a second-order ODE

I think that's just a ODE with non constant coeffecients, since expanding yields

P(t)x''+P'(t)x'+q(t)x=0

You may be able to solve this with power series if P and q fit them.
Non linear differential equations rarely have closed form solutions.
But that's okay, we have computers
 Of course, when I say "There is no general solution", I mean "No general analytical solution espressed on a closed form". Obviously, in some particular cases, with some particular forms of functions p(t) and q(t), the solutions might be known on closed form, and/or be expressed as infinite series. Even more generally the solutions can be accurately approached thanks to numerical methods.