I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions?

$$(1-t^2)u_{tt}-tu_t+\left[ n \beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$

$C$ is a free parameter. So if you know a function that would fulfill this equation only for particular $C$, this would be perfectly fine. I am interested in its solutions on $[-1,1]$.

I should include the motivation/reference here: Physics (Quantum Mechanics) and you might want to see this great answer that gives us first hints about the structure of the solution (due to O.L.) https://math.stackexchange.com/questions/780591/how-to-make-a-smart-guess-for-this-ode

The problem is that the approach taken here does not offer an analytical representation of the solution. I suspect that the solutions form a nice orthogonal basis of $L^2[-1,1]$, but was incapable of constructing them by recursion or explicit representation.