Q.Assume that the rate at which radioactive nuclei decay is proportional to the number of such nuclei that are present in a given sample. In a certain sample $10\%$ of the original number of radioactive nuclei have undergone disintegration in a period of $100$ years. What percentage of the original radioactive nuclei will remain after $1000$ years?

A single class of first-order differential equation models a wide range of real phenomena: whenever a quantity's instantaneous rate of change depends on the quantity itself (and possibly time), the governing equation has the form $\dfrac{dx}{dt}=f(x,t)$.

Population growth (Malthusian law). If a population $x(t)$ grows at a rate proportional to its current size, $\dfrac{dx}{dt}=kx$ with $k>0$. This is separable: $\dfrac{dx}{x}=k\,dt\Rightarrow\ln x=kt+C\Rightarrow x=x_0e^{kt}$, where $x_0$ is the population at $t=0$. Doubling/tripling-time problems fix $k$ from one data point and then answer a second question using the same $k$.

Radioactive decay. The same law with a negative rate constant, $\dfrac{dA}{dt}=-kA\ (k>0)$, models decay: $A(t)=A_0e^{-kt}$. The half-life is the time $t_{1/2}$ at which $A=\dfrac{A_0}{2}$, found by solving $\tfrac12=e^{-kt_{1/2}}$.

Newton's law of cooling/warming. The rate at which a body's temperature $T(t)$ changes is proportional to the difference between $T$ and the constant ambient temperature $T_m$:

Separating and integrating gives $T-T_m=Ce^{kt}$; the constant $C$ is fixed by the temperature at $t=0$, and $k$ is fixed from one later reading, after which the model predicts the temperature (or the time to reach a target temperature) at any other instant.

Mixture (tank) problems. For a substance flowing into and out of a well-stirred tank, the amount $x(t)$ present obeys

$\dfrac{dx}{dt}=\text{IN}-\text{OUT},$ …