Model unrestricted growth and decay with $\frac{dN}{dt} = k N$ and solve the resulting separable differential equation

What this dot point is asking

NESA wants you to recognise scenarios that follow the unrestricted exponential growth or decay model $\frac{dN}{dt} = k N$, write down the general solution, use given data points to find the constants, and solve problems involving doubling, halving and prediction.

The answer

The model

If a quantity $N(t)$ grows or decays at a rate proportional to itself,

$k > 0$: growth. $k < 0$: decay. The constant $k$ is the proportionality constant.

General solution

Separate variables:

Integrate:

where $N_0 = N(0)$ is the initial value.

Finding $k$ from data

Given $N(t_1) = N_1$ and $N(0) = N_0$:

For decay (where $N_1 < N_0$), $k < 0$.

Doubling time and half-life

Doubling time for growth: solve $2 N_0 = N_0 e^{k T_d}$ to get $T_d = \frac{\ln 2}{k}$.

Half-life for decay: solve $\frac{1}{2} N_0 = N_0 e^{k T_h}$ to get $T_h = \frac{\ln \frac{1}{2}}{k} = -\frac{\ln 2}{k}$ (positive when $k < 0$).

Both are independent of $N_0$, which is the defining feature of exponential growth and decay.

Continuous compound interest

A bank account earning interest at a continuous rate $r$ satisfies $\frac{dA}{dt} = r A$, so $A(t) = A_0 e^{r t}$. This is the same model.

Populations with capped growth (logistic)

For populations, the model $\frac{dN}{dt} = k N (M - N)$ caps growth at carrying capacity $M$. Extension 1 mostly stays with unrestricted growth; logistic growth appears in Extension 2 and in some applications.

The simpler model for population control in Extension 1 is

which models cooling or approach to equilibrium. Its solution is $N = N_\infty + (N_0 - N_\infty) e^{-k t}$ (Newton's law of cooling form). See the separable differential equations dot point for derivation.