Differential equations are equations with derivatives. They represent rates of change.
They model physical and economic systems, such as population growth, heat transfer, and compound interest.
P
represents the population size in people, with
t
as years:
\frac{dP}{dt} = 0.02P
The population is increasing by 2% each year.
Solving a differential equation is generally to find a function which satisfies it. For this example, we are finding
P(t)
:
P(t) = ke^{0.02t}
Where
k
is the initial population. We can check this solution through differentiation:
\begin{align*}
P(t) &= ke^{0.02t} \\ \text{} \\
\frac{d}{dt}P(t) &= \frac{d}{dt}ke^{0.02t} \\ \text{} \\
\frac{dP}{dt}&= 0.02ke^{0.02t} \\ \text{} \\
\frac{dP}{dt}&= 0.02P
\end{align*}
As with antiderivates, the last missing piece is finding the constant. For example, if the population is 310 at
t=100
years, then:
\begin{align*}
P(t) &= ke^{0.02t} \\ \text{} \\
310 &= ke^{0.02(100)} \\ \text{} \\
\frac{310}{e^{2}} &= k \\ \text{} \\
k &= 42 \\ \text{} \\
P(t) &= 42e^{0.02t}
\end{align*}
The constant
k
is 42, as
P(100) = 310 = 42e^{0.02(100)}
.