Problem

(a) Express

$$\frac{1}{(1+3x)(1-x)}$$

in partial fractions.

(b) Hence find the solution of the differential equation

$$(1+3x)(1-x)\frac{dy}{dx}=\tan y\qquad -\frac13<x\le \frac12$$

for which $x=\dfrac12$ when $y=\dfrac{\pi}{2}$

Give your answer in the form $\sin^n y=\mathrm{f}(x)$ where $n$ is an integer to be found.