Why does one often use the following notation for differential equation:
y'=f(t)y (this is just a particular example) ?
What bothers me with this notation, which I have encountered in countless textbooks is, that one mixes in this notation the symbols which denote functions ($y',y$) with those that denote function values ($f(t)$). Shouldn't the above be written either as y'=f \cdot y (where the multiplication is pointwise understood) or as $\forall t\in I:\ y'(t)=f(t)\cdot y(t)$where $I$ is for example an interval - meaning either just on the level of linking functions with functions or just on the level of linking functions values with functions values ?
How come, that in the subject of differential equations sloppy (even bad/confusing) notation is more the norm than the exception ?
Side question: In a course I read a while ago, someone defined a function t(y(x))=y(x)+y'(x). My question is: Is this even correct?
Because one can't just define a function like that; either one defines directly a function $u$ as u(x)=y(x)+y'(x) or either one defines $t$ and the composes $t$ with $y$. But in the last case (which was as one that was meant in the course) how should $t$ look ? One can't define a function $t:I\subseteq \mathbb{R} \rightarrow \mathbb{R}$, as far as I know, such that t(y(x))=y(x)+y'(x) for all suitable differentiable functions $y$.