I have a Schrödinger-like wave equation,
$$ \frac{d^2\psi}{dx^2} +\left[ \omega^2 + Q_0 + \frac{Q'''}{3!}(x-x_0)^3\right]\psi = 0 $$ from which we note there is one pole in $x \to \infty$. Following this article, we can convert any ODE with less than three ordinary singular points into a hypergeometric equation, $$ z(z-1)F''+[(a+b+1)z-c]F'+ abF = 0 \,\, . $$
To perform this conversion, I think it's necessary to eliminate the term proportional to $F'$. So my question are:
- Is it really possible to convert the wave equation into the Hypergeometric one? Why?
- What's the mechanism to perform this conversion (change of variables, new parametrization)?