Problem

(a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that

$$\sinh 2x=2\sinh x\cosh x.$$

(b) Using the substitution $u=\sinh x$, find

$$\int\sinh^2 2x\cosh x\,\mathrm{d}x.$$

(c) Find the particular solution of the differential equation

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

given that $y=4$ when $x=0$. Give your answer in the form $y=f(x)$.