$$y''-xy=0$$

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

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

$$xy''+(k+1-x)y'+(n-k)y=0$$

$$y''-2xy'+2ny=0$$

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

$$y''+(\lambda-x^2)y=0$$

$$ \frac{1}{\sin\theta}\frac{d}{d\theta} \left(\sin\theta\frac{d}{d\theta}\right)f(\theta) +\lambda-\frac{m^2}{\sin\theta}f(\theta)=0 $$