after applying the variable separation method to a 2D problem, suppose, I have the following equation
$a\frac{f(x)}{\frac{d\{f(x)\}}{dx}}+b\frac{g(y)}{\frac{d\{g(y)\}}{dy}}=c$ where $a,b,c$ are constants and a 2D function $F(x,y)$ has been decomposed as $F(x,y)=f(x)g(y)$. Since the left hand side is equals to a constant, can I write
$a\frac{f(x)}{\frac{d\{f(x)\}}{dx}}=P$ and $b\frac{g(y)}{\frac{d\{g(y)\}}{dy}}=Q$
where $P$ and $Q$ are constant with $P\neq Q$
Yes, this is true.
You have two functions $r$ and $s$, such that $r(x)+s(y) = c$, where $$r(x) = a\frac{f(x)}{f'(x)},$$ and $$s(y) = b\frac{g(y)}{g'(y)}.$$
Let $y_1\neq y_2$, and fix some $x_0$. Then $$r(x_0)+s(y_1) = c,$$ $$r(x_0)+s(y_2) = c.$$ Subtract the first equation from the second to obtain $$s(y_2)-s(y_1) = 0.$$ This means that $s(y_2)=s(y_1)$. We chose arbitrary $y_1$ and $y_2$, so the value of $s(y)$ is the same for all $y$, meaning that $s$ is a constant function. Likewise for $r$.