CAIE A-Level Mathematics A2 3.8.1 Differential Equations Questions

Practise forming differential equations from rate relationships and eliminating constants.

Exam points

form differential equations from given relationships
eliminate arbitrary constants from families of curves
use rate models to derive first- and higher-order equations

[Maximum number: 2]

A large plantation of area $20 \mathrm{~km}^{2}$ is becoming infected with a plant disease. At time t years the area infected is $x \mathrm{~km}^{2}$ and the rate of increase of x is proportional to the ratio of the area infected to the area not yet infected.

When t=0, x=1 and $\frac{\mathrm{d} x}{\mathrm{~d} t}=1$ .

Show that x and t satisfy the differential equation

\frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{19 x}{20-x} .

State or imply equation of the form $\frac{\mathrm{d} x}{\mathrm{~d} t}=k \frac{x}{20-x}$
Obtain k=19
AG

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CAIE A-Level Mathematics A2 3.8.1 Differential Equations Questions question 1