I don't see any typo no, that's the notation we use for Ordinary Differential Equations.

Differentiating that will give you the $\displaystyle \frac{-1}{x^2}y$ yes, but what I think I'm meaning to say is, how do you jump from the $\displaystyle \frac{dy}{dx}$ to $\displaystyle \frac{d}{dx}$ what is the difference between them.

And if you do as you say and differentiate w.r.t. to

the expression

? How can this still be the same as the line up above it? Surely that would make it: $\displaystyle \frac{-1}{x^2}y=e^2x$ and not

Or am I completely missing something here?