Problem

(a) Show that

$$\frac{\mathrm{d}}{\mathrm{d}x}\big(\ln(\tanh x)\big) = 2\operatorname{cosech}2x.$$

(b) Find the solution of the differential equation

$$\sinh 2x\frac{\mathrm{d}y}{\mathrm{d}x} + 2y = \sinh 2x$$

for which $y = 5$ when $x = \ln 2$. Give your answer in an exact form.