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]

In a field there are 300 plants of a certain species, all of which can be infected by a particular disease. At time t after the first plant is infected there are x infected plants. The rate of change of x is proportional to the product of the number of plants infected and the number of plants that are not yet infected. The variables x and t are treated as continuous, and it is given that $\frac{\mathrm{d} x}{\mathrm{~d} t}=0.2$ and x=1 when t=0.

Show that x and t satisfy the differential equation

1495 \frac{\mathrm{~d} x}{\mathrm{~d} t}=x(300-x) .

State or imply equation of the form $\frac{\mathrm{d} x}{\mathrm{~d} t}=k x(300-x)$ and use $\frac{\mathrm{d} x}{\mathrm{~d} t}=0.2$ and x=1
M0 for verification.

Obtain $k=\frac{1}{1495}$ and rearrange to the given answer
$1495 \frac{\mathrm{~d} x}{\mathrm{~d} t}=x(300-x)$ .

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