(a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that
\(\sinh 2 x=2 \sinh x \cosh x .\)
(b) Using the substitution \(u=\sinh x\), find \(\int \sinh ^{2} 2 x \cosh x \mathrm{~d} x\).

(c) Find the particular solution of the differential equation
\(\frac{\mathrm{d} y}{\mathrm{~d} x}+y \tanh x=\sinh ^{2} 2 x,\)
given that \(y=4\) when \(x=0\). Give your answer in the form \(y=\mathrm{f}(x)\).