Sloppy notation $ y'=f(t)y$ for differential equation $ y'(t)=f(t)y(t)$ 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 $y$.
 A: My answer, in short, is that you are right on all mathematical counts. 
In your first example, if $y$ and $f$ are functions defined on $\mathbb R$ or on suitable subsets of $\mathbb R$, to write $y'=f(t)y$ is absurd. One should write either that $y'=fy$ (a relation between two functions, namely $y'$ and $fy$) or that $y'(t)=f(t)y(t)$ (a relation between two real numbers, namely $y'(t)$ and $y(t)f(t)$) for every suitable $t$.
Hence, to say something like $f(t)$ is a function of $t$ is literally meaningless since $f(t)$ is a number and not a function.
Some sorry consequences of this confusion are manifest in your second example (as you noticed), since the expression $t(y(x))=y(x)+y'(x)$ can only mean that $y$, $y'$ and $t$ are functions and that $t$ is defined as folllows. For every $z$ in the image set of the function $y$, either there exists a unique point $x$ such that $y(x)=z$ and then one defines the image of the function $t$ at the point $z$ as $t(z)=z+y'(x)$ for this unique $x$, or there are several points $x$ such that $z=y(x)$ but these all have the same image by $y'$ hence one can use any of them to define $t(z)$. All this breaks down if $y'(x_1)\ne y'(x_2)$ for two points $x_1$ and $x_2$ such that $y(x_1)=y(x_2)$.
The only count on which I differ with you, or at least, on which I beg to suspend my approval, is when you write that in the subject of differential equations sloppy (even bad/confusing) notation is more the norm than the exception. This is too broad and sweeping a statement for my taste, unless you can back it up with some solid evidence (and if you try to do that (that is, muster some evidence), you will soon realize that the subject of differential equations is treated at very different levels of rigor, depending on the intended audience).
