Say you are given a simple differential equation $dy/dx=k*f(x)*g(y)$ where $k$ is a constant, $f(x)$ is some function of $x$, and $g(y)$ is some function of $y$. Am I allowed to move the constant $k$ to the other side of the equation while applying separation of variables such that the equation becomes $dy/(k*g(y))=f(x)dx$ and then integrate each side with respect to the corresponding variable? I am asking this because I did something like this on a math competition today and got the wrong answer as a result, and when I kept k on its original side, I got the right answer. Keep in mind the problem was an initial value problem, so that may have an effect, but I'm not sure. So I am allowed to move constants during separation of variables, and why or why not? Thanks in advance.
2026-03-24 22:01:28.1774389688
Moving constants around during separation of variables
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