Example 6. Suppose that at a constant temperature, the dissolution rate of a solid in a liquid is proportional to the amount of this substance that can still dissolve in the liquid until it becomes saturated (it is assumed that the substances entering the solution do not chemically interact with each other, and the solution is far from saturation, otherwise the linear law for the dissolution rate would not apply). Find the dependence of the amount of dissolved substance on time.
Solution. Let $P$ be the amount of substance that gives a saturated solution, and $x$ be the amount of substance already dissolved. Then we obtain the differential equation

$$
\frac{d x}{d t}=k(P-x)
$$

where $k$ is a proportionality coefficient known from experience, and $t$ is time. Separating variables, we find

$$
\frac{d x}{P-x}=k d t
$$

Integrating, we get

$$
\ln |x-P|=\ln C-k t, \quad \text { from which } \quad x=P+C e^{-k t}
$$

At the initial moment $t=0$, we have $x=0$, so $C=-P$, hence finally

$$
x=P\left(1-e^{-k t}\right)
$$