(i)[1]
Show, by differentiating, that $\frac{d}{d\theta}(\tan^2\theta) = \frac{2\tan\theta}{\cos^2\theta}$.
(ii)[7]
Solve the differential equation and find $x$ when $\theta = \frac{1}{3}\pi$, giving your result to $3$ significant figures.
Mathematics 9709 · AS & A Level · Differential equations
Differential equations — practice question
Variables $x$ and $\theta$ are related by the differential equation $x\cos^2\theta\,\frac{dx}{d\theta} = 2\tan\theta + 1$, for $0 \le \theta < \frac{1}{2}\pi$ and $x > 0$. It is given that $x = 1$ when $\theta = \frac{1}{4}\pi$.