Separation of variables can be explained by the chain rule. If your differential equation is [tex]\frac{dy}{dx} = f(x)/g(y) [/tex] (in your example, f(x)=k and g(y)=1/y) then we can write
[tex] g(y) \frac{dy}{dx} dx = f(x)[/tex]
These are both functions of x (because g(y) = g(y(x))). So integrate them both
[tex] \int g(y) \frac{dy}{dx} dx = \int f(x)dx[/tex]
and the result follows from noting that
[tex] \int g(y(x)) \frac{dy}{dx} dx = \int g(y)dy[/tex]
by doing integration by parts (with the substitution y=y(x)).
Then you can consider the act of separating the variables as just a mnemonic shortcut to avoid a lot of notation
