How to solve or transform this differential equation (to a classical form)?

Question

We have the following ODE $$\left(\frac{d^2}{{dx}^2}+\frac{1}{x}\frac{d}{dx}-\frac{\alpha^2}{x^2}-\beta^2x^2+\gamma\right)y(x)=0$$ with $x>0$ and $\alpha,\beta,\gamma\in\mathbb{R}$. Is it possible to transform it to some classical form and thus relate its solution to some corresponding special functions (like Bessel function or so)? Thanks in advance.

Answer 1

With $t = \beta x^2$ and $u(t) = x y(x)$, you get a Whittaker differential equation.

$$ u''(t) + \left(\frac{1-\alpha^2}{4t^2} + \frac{\gamma}{4\beta t} - \frac{1}{4}\right) u(t) = 0 $$