How Do You Set Up a Differential Equation From a Real-World Context?

You set up a differential equation by defining the changing quantity and independent variable, translating each stated rate into mathematical terms, and combining those rates into a net rate of change. This is HL-only content in AHL 5.14, commonly tested in extended-response questions.

The Modelling Process

Let the changing quantity be $P$, and let time be $t$. The derivative $\frac{dP}{dt}$ represents how quickly $P$ changes with time.

Use this structure:

$$\frac{dP}{dt}=\text{rate in}-\text{rate out}$$

Before writing the equation, identify the units. If $P$ is measured in fish and $t$ in years, every term in $\frac{dP}{dt}$ must have units of fish per year. This dimensional consistency check often reveals a missing factor of $P$.

Translate key phrases carefully:

Worked Example

A fish population $P$ grows continuously at a rate proportional to its size. The proportional growth rate is $8\%$ per year, while $120$ fish are removed each year.

The natural growth rate is $0.08P$, and the removal rate is $120$. Therefore,

$$\frac{dP}{dt}=0.08P-120.$$

If the initial population is $2000$, include the initial condition

$$P(0)=2000.$$

The equation models continuous change and assumes that the percentage growth rate and annual removal rate remain constant.

Setting the net rate to zero gives an equilibrium: $0.08P-120=0$, so $P=1500$. The model predicts growth above $1500$ and decline below it.

A common misconception is to write $\frac{dP}{dt}=0.08-120$. The percentage rate must multiply the current population, so the correct term is $0.08P$.

Exam Technique

For an IB response, define variables, show how every phrase produces a term, include units and any initial condition, and check whether each term increases or decreases the quantity. Do not begin separating variables until the differential equation has been correctly established.