$$y''-xy=0$$

$$x^2y''+xy'+(x^2-n^2)y=0$$

$$\frac{d}{dx}\left\{(1-x^2)\frac{dy}{dx}\right\}+n(n+1)y=0$$

$$x(1-x)y''+(\gamma-(\alpha+\beta+1)x)y'-\alpha\beta y=0$$