How do we know which component belongs to which part in a separable differential equation Take for instance dP/dt = kP
We get after separating: dP/P = kdt, but why shouldn't it be 
dP/kP = dt instead, mathematically it doesn't make sense to say that k must belong absolutely to the right hand side of the equation.
 A: Who said $k$ has to be on the right hand side? It is a constant, so it can be on either side. The important point is that anything in terms of $P$ is on the side with $dP$ and everything in terms of $t$ is on the side with $dt$. Since a constant is independent of $P$ and $t$, it may go on either side.
Let's solve $\dfrac{dP}{dt} = kP$ both ways to illustrate why it doesn't matter.


*

*First we separate the equation as
$$\frac{dP}{P} = k \, dt.$$
Integrating both sides, we get
$$\ln |P| = kt + C.$$
Exponentiating both sides, we get the solution
$$P = e^{kt + C} = Ae^{kt},$$
where $A$ is an arbitrary constant.

*Now separate the equation as
$$\frac{dP}{kP} = dt.$$
Integrating both sides, we get
$$\frac{1}{k} \ln |P| = t + C.$$
Multiplying both side by $k$ results in
$$\ln |P| = kt + C',$$
where $C' = kC$. Since $C$ is an arbitrary constant, this is an irrelevant detail. Now exponentiating both sides gives
$$P = e^{kt + C'} = Ae^{kt},$$
where $A$ is an arbitrary constant.
So both methods give the same general solution. The reason for this is that we can factor out a constant from an integral, so the side that the constant is on doesn't affect the integration.
The reason every book will leave the $k$ on the right-hand side for this problem is because we want a solution for $P$; we'll just have to move $k$ back to the right-hand side after integrating if we start with it on the left-hand side with $P$.
