Mathematics 9709 · AS & A Level · Differential equations
Differential equations — practice question
The variables $x$ and $\theta$ satisfy the differential equation $x \sin^2 \theta \, \frac{dx}{d\theta} = \tan^2 \theta - 2 \cot \theta$, for $0 < \theta < \frac{1}{2}\pi$ and $x > 0$. It is also given that $x = 2$ when $\theta = \frac{1}{4}\pi$.
(a)[1]
Show that $\frac{d}{d\theta}(\cot^2 \theta) = -\frac{2 \cot \theta}{\sin^2 \theta}$. (You may assume, without proof, that $\frac{d}{d\theta}(\cot \theta) = -\cosec^2 \theta$.)
(b)[7]
Solve the differential equation and determine the value of $x$ when $\theta = \frac{1}{6}\pi$.
Worked solution & mark scheme
This 8-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Show enough working to justify the statement given” …