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Why are some functions denoted by capital letters? For example, an ODE is represented by the equation $F(t, y, y',\ldots, y^n) = 0$. Here, what I understand is that F is actually a multivariate function. Similarly, for $dy/dx=f(x,y)$ if we consider $f(x,y)=-M(x,y)/N(x,y)$ we get separable equation. Why represent $M$ and $N$ with capital letters? Are they different from the functions represented by small letters?

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No; there is no inherent difference in the symbol you use, unless this is a local convention by the author. We could call the function $x$ and $y$ and the variables $f$ and $g$, it wouldn't make a difference as far as what the function is. – Arturo Magidin Jun 27 '12 at 4:42
....but you should always treat $F$ and $f$ as two different letters. Often they refer to two different things in the same problem. – Michael Hardy Jun 27 '12 at 4:56
Bold letters $\ne$ CAPITAL letters. I've edited the question to fix it. – Rahul Jun 27 '12 at 5:22
@Arturo, it would make an enormous difference as far as readability is concerned. $dy/dx=f(x,y)$ is instantly comprehensible to anyone au fait with differential equations; $df/dg=x(f,g)$ would make me wonder what the author had been smoking. – Gerry Myerson Jun 27 '12 at 5:57
@Gerry: Yes, of course; the joke someone proposed once was to just use "the next letter" any time you needed a symbol, and as an example wrote the definition of continuity that way. It was nigh unreadable. But the point is that it's just symbols, and while we tend to select certain symbols for context, that context is entirely created "outside" the mathematical objects in question. We are used to certain symbols "signaling" certain kinds of objects, but that's mere social convention, not a mathematical difference inherent in the chosen symbols (though I know I'm preaching to the choir). – Arturo Magidin Jun 27 '12 at 15:59

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