Model and solve practical situations with first-order differential equations, including exponential growth and decay, Newton's law of cooling and the logistic equation, and interpret the long-term behaviour of solutions

What this dot point is asking

QCAA wants you to recognise the standard first-order models, set up the differential equation from a worded scenario, solve it (usually by separation of variables), apply initial conditions, and interpret long-term behaviour. The three core models are exponential growth and decay, Newton's law of cooling, and the logistic equation, each with a characteristic solution and limiting value.

The answer

Exponential growth and decay

When a quantity changes at a rate proportional to its current size,

If $k > 0$ the quantity grows without bound; if $k < 0$ it decays toward zero. This models populations, radioactive decay and continuously compounded interest. The half-life or doubling time follows from $e^{kt}$.

Newton's law of cooling

The temperature $T$ of an object changes at a rate proportional to the difference from the ambient temperature $T_s$:

Separating variables gives the solution

where $T_0$ is the initial temperature. As $t \to \infty$, $e^{-kt} \to 0$, so $T \to T_s$: the object approaches the surrounding temperature. The ambient temperature $T_s$ is the equilibrium.

The logistic equation

Population growth limited by resources is modelled by

where $M$ is the carrying capacity. When $P$ is small the bracket is near $1$ and growth is nearly exponential; as $P$ approaches $M$ the bracket approaches $0$ and growth slows. The solution is the S-shaped logistic curve

Long-term behaviour and equilibria

Each model has a limiting value. Exponential growth has none (it diverges); decay tends to $0$; cooling tends to $T_s$; the logistic model tends to its carrying capacity $M$. Equilibria are the constant solutions where the rate is zero: for the logistic equation, $\frac{dP}{dt} = 0$ at $P = 0$ (unstable) and $P = M$ (stable).

Setting up from a scenario

Translate phrases into the equation. "Rate proportional to amount" gives $\frac{dN}{dt} = kN$. "Rate proportional to the temperature difference" gives cooling. "Growth limited by a maximum sustainable level" signals the logistic model with $M$ that maximum. Then separate variables and integrate, applying the initial condition to find the constant.

Interpreting the constant $k$

The constant $k$ sets the timescale: a larger $|k|$ means faster change. Solve for $k$ using a second data point (for example a known value at a later time) when the scenario provides one.