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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 $y$.

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Take $y'=f(t)y$ as an abbreviation for "$y'(t)=f(t)y(t)$ for all $t$." The unabbreviated version is not all that attractive. –  André Nicolas Oct 9 '11 at 16:16
I think it's basically a physics convention where everything is secretly a function of $t$, except when it's stated or implied that it isn't. So "$f(t)$" implies that the functional relation $f$ itself is always the same (and presumably is given by an explicit expression), but everything that's not written with a $\cdot(t)$ is implicitly time-varying. –  Henning Makholm Oct 9 '11 at 16:19
... in other words, the difference between "$f(t)$" and "$y$" is a mnemonic device to help remember that the relation between $t$ and $f(t)$ is assumed to be known already, whereas the relation between $t$ and $y$ is unknown. One way to formalize this notation partially is to declare that every letter belongs to a space of functions of a hidden unnamed variable $\tau$, such that $t$ in particular stands for the identity function $t(\tau)=\tau$, and things like $f$ are implicitly supposed to be constant functions of $\tau$. –  Henning Makholm Oct 9 '11 at 16:34
@Henning Makholm This seems to be interesting. But I didn't understood what you meant with "...implies that the functional relation f itself is always the same..."It sounded like you implied that $f$ was a constant function: $f(t)=c\ \ \forall t\in I$... –  temo Oct 9 '11 at 16:38
I suppose you haven't studied the calculus of variations yet. :) –  KCd Oct 9 '11 at 19:33
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1 Answer 1

up vote 7 down vote accepted

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).

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Could you maybe give me an example of a book (or of books - the more I can choose from, the better :) in which the subject of differential equations is treated very rigorously (even that means prerequisites of functional analysis or higher analysis!) ? (My very question was posed out of the reason that every differential equation book I looked into, I found this kind of sloppiness I described.) –  temo Oct 14 '11 at 17:44
Dieudonné's Éléments d'analyse were translated in English as a Treatise on analysis, whose first volume, called Foundations of modern analysis, might contain an example of what you are looking for (chapter X). Here is a sample of the French edition: *Une application différentiable $u$ d'une boule ouverte $J\subset I$ dans $H$ est appelée une solution de l'équation différentielle $x'=f(t,x)$ si, pour tout $t\in I$, on a $u'(t)=f(t,u(t))$* (and naturally $f$, $I$ and $H$ were properly introduced some lines before). Having said that, .../... –  Did Oct 17 '11 at 14:23
.../... I should add that your (justified) search for rigor in notations should not prevent you to peruse also some texts with a more relaxed style of exposition, at least on subjects you feel you master enough and provided you take these texts with a grain of salt (as we should always do). The example of Arnol'd comes to mind here. –  Did Oct 17 '11 at 14:24
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