Separation of variables - "formal" notation? When using separation of variables technique to solve differential equations, I sometimes have both f(x) and g(y) on the right side, and then I divide by g(y) to separate them.... but how can I explain formally that I am allowed to do this? 
You could say that g(y) does not equal 0.... but often, g(y) = 0 IS a solution to a differential equation. So what then? 
It's not a question directly related to an assignment, it's just that my teacher wants us to be very formal and explain every move we make instead of just copying down the methods from the book. 
 A: Note that dividing by by $g(y)$ is problematic as long as $g(a) = 0$ anywhere, not just the particular case of $g(y)$ being identically zero.
This means that the result of your calculation will only be guaranteed to hold on the regions where $g(y)$ is everywhere nonzero.
However, that's usually a lot of information, and it often doesn't take much work to fill in the gaps in that information or separately work out the remaining cases that this doesn't tell you anything about.
As a very simple example of how that kind of argument can go, if you know that $f(x)$ is everywhere differentiable and that on $x > 0$, you must have $f(x) = ax$ for some constant $a$, and that on $x < 0$ you must have $f(x) = bx$ for some constant $b$, then by checking the one-sided limits of $f$ and $f'$, you can infer that for some constant $c$, you have $f(x) = cx$ everywhere.
(note that usually in such a situation, you have no a priori reason to expect the constant to be the same on $x>0$ and $x<0$, which is why I used different variables in those cases prior to the determination that they have to be the same)
