How do I know when it is appropriate to integrate a differential equation?

Question

I am beginning a differential equation textbook, and in the first problem section I encountered a question regarding the change in a population of mice. I am given that $\frac{dP}{dt} = kP^2$, $P(0) = 2$, and that $\frac{dP}{dt} = 1$ when $P = 10$. It is simple to find the value of $k$, but I am struggling to figure out at what time the population of mice is equal to 100. I have found a solution online, and it integrates the given differential equation $$\int \frac{dP}{dt} = kP^2 \rightarrow \frac{-1}{P} + C = kt$$

I understand the mechanics of the integral, but I'm confused as to why I can integrate at all. Is it implied that P is just a value and not a function? It seems here that they are assuming that $P(t)$ is a variable and not a function. Couldn't $P$ be a function of t, making it inappropriate to integrate?

Answer 1

The separation of variables is being used here: $$\frac{dP}{dt} = kP^2$$ $$\frac{dP}{P^2} = kdt$$
$$\int{\frac{dP}{P^2}} = \int{kdt}$$ Since left hand side is using $dP$ and $P$, we can integrate as if $P$ was an independent variable. Finally, $$-\frac{1}{P} = kt +c$$

Answer 2

In posting this question, I realized the answer. Say you have a function $g(x)$. It would be difficult (impossible?) to evaluate the integral $\int g(x)\ dx$ without more information about $g(x)$, however $\int g(x)\ dg(x) = \frac{1}{2} g(x)^2$. Since we are evaluating the integral on with respect to $P$ and not to $t$ (for the lefthand side of my provided integral), it is correct to evaluate it as so.