Problem

(a) Find the derivative with respect to $y$ of

$$\frac{1}{(1+2\ln y)^2}$$

(b) Hence find a general solution to the differential equation

$$3\cosec(2x)\frac{dy}{dx}=y(1+2\ln y)^3 \qquad y>0 \qquad -\frac{\pi}{2}<x<\frac{\pi}{2}$$

(c) Show that the particular solution of this differential equation for which $y=1$ at $x=\dfrac{\pi}{6}$ is given by

$$y=e^{A\sec x-\frac{1}{2}}$$

where $A$ is an irrational number to be found.