Exponential Growth and Decay

Definition For a positive differentiable quantity $y(t)$, the relative growth rate at time $t$ is

The quantity has constant relative growth rate $k$ on an interval $I$ if $y'(t) = k\,y(t)$ for every $t$ in $I$.

Definition The differential equation

with a constant $k$, is the law of natural growth when $k > 0$ and of natural decay when $k < 0$. On an interval containing $0$, the differentiable solutions are exactly the functions

To see that there are no others, note that every solution satisfies

so $y(t)e^{-kt}$ is constant on the interval and equals $y(0)$.

Definition A population $P(t)$ with constant relative growth rate $k > 0$ satisfies $P' = kP$, so $P(t) = P_0 e^{kt}$ where $P_0 = P(0)$. Two measurements at different times determine both $P_0$ and $k$. The model fits as long as food and space do not limit growth.

Definition The mass $m(t)$ of a radioactive substance satisfies $m' = -rm$ for a constant $r > 0$, so $m(t) = m_0 e^{-rt}$. The half-life $T$ is the time needed for any given amount to fall to half of itself. From $e^{-rT} = \frac{1}{2}$,

The half-life does not depend on the starting amount or the starting time.

Definition Newton’s law of cooling says that the temperature $T(t)$ of an object in surroundings held at constant temperature $T_s$ changes at a rate proportional to the difference,

Applying the law of natural decay to $y = T - T_s$ gives

Definition An amount $A_0$ invested at annual interest rate $r$ compounded $n$ times a year grows after $t$ years to

For fixed $t$, as $n \to \infty$ this tends to

which is called continuous compounding. It satisfies $A'(t) = rA(t)$, the law of natural growth with $k = r$.

Definition A function $y$ solves the initial value problem $y' = F(t, y)$, $y(t_0) = y_0$ on an interval $I$ containing $t_0$ if $y$ is differentiable on $I$, $y'(t) = F(t, y(t))$ for every $t$ in $I$, and $y(t_0) = y_0$. Checking a proposed answer means verifying both conditions directly.