CAIE A-Level Mathematics 3.8.1 Forming Differential Equations

Practise converting proportionality and inflow–outflow statements into differential equations and using volume–dimension relationships with the chain rule to obtain the required…

Exam points

introduce a positive proportionality constant with the sign fixed by growth or decay
write the net physical rate as inflow minus outflow in the stated variables
use dV/dt = (dV/dx)(dx/dt) and given data to determine the constant and final model

[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 $\frac{\mathrm{d} x}{\mathrm{~d} t}=k \frac{x}{20-x}$
M1
Obtain k=19
A1
AG

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3.8.1—Differential equations question 1