Sloppy notation $ y'=f(t)y$ for differential equation $ y'(t)=f(t)y(t)$

Solution 1:

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).