Differential Equations · Paper 1 — Free Maths AI HL Practice Questions

The population of a small island is currently $5000$. The rate of change of the population $P$ (measured in thousands) with respect to time $t$ (measured in years) is modelled by the differential equation $$\frac{dP}{dt} = 0.02P$$

A population of rabbits on an island is modelled by the differential equation $$\frac{dP}{dt} = \frac{1}{200}P(1000 - P)$$ where $P$ is the population size at time $t$ years. Initially, there are $200$ rabbits on the island.

A pollutant is being filtered from a lake. The volume of pollutant, $V$ (measured in cubic metres), in the lake at time $t$ (measured in days) satisfies the differential equation $$\frac{dV}{dt} = -\frac{V}{t+1} + 2, \quad t \geq 0$$ Initially, when $t = 0$, there are $10\,\text{m}^3$ of pollutant in the lake.