I am confused about the meaning of the unknown function $\displaystyle y$ in a differential equations.

For example, the Sturm-Liouville equation has the form $\displaystyle \frac{d}{dx} \left( p(x) \frac{dy}{dx} \right) + (q(x) + \lambda r(x) ) y = 0$. I notice that the unknown function $\displaystyle y$ is written without its variable $\displaystyle x$, i.e. not written as $\displaystyle y(x)$, whereas the functions $\displaystyle p(x)$, $\displaystyle q(x)$ and $\displaystyle r(x)$ are written with variable $\displaystyle x$.

The following rearrangement of equation is from my lecture notes about orthogonality of eigenfunctions of the Sturm-Liouville equation.

$\displaystyle (\lambda_m - \lambda_n) r(x) \phi_n \phi_m = \phi_m \frac{d}{dx} \left( p(x) \frac{d\phi_n}{dx} \right) - \phi_n \frac{d}{dx} \left( p(x) \frac{d \phi_m}{dx} \right) $

$\displaystyle (\lambda_m - \lambda_n) r(x) \phi_n \phi_m = \frac{d}{dx} \left( p(x) \frac{d \phi_n}{dx} \phi_m - p(x) \frac{d \phi_m}{dx} \phi_n \right) $

Why it is OK to move an unknown function $\displaystyle \phi_m$ of variable $\displaystyle x$ into a derivative operator $\displaystyle \frac{d}{dx}$ by treating it like a constant? I am wondering if I have missed some very basic points about differential equations. Can someone please explain to my why the above rearrangement is possible and when one would write an unknown function as $\displaystyle y$ and when one would write an unknown function as $\displaystyle y(x)$ in an ODE?