CAIE A-Level Mathematics 3.8.2 Separable Differential Equations

Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise separating first-order differential equations, integrating both sides and applying an initial condition.

How this is tested

move all y terms with dy and all x terms with dx before integrating
use the required integration technique and include an arbitrary constant
apply the initial condition, then rearrange into the requested relation or explicit form

[Maximum number: 8]

The variables x and y satisfy the differential equation

\sin 4 y \frac{\mathrm{~d} y}{\mathrm{~d} x}=x \sin 2 y \sin 3 x .

It is given that $y=\frac{1}{12} \pi$ when $x=\frac{1}{2} \pi$ .

Solve the differential equation, obtaining a relation between x and y.