Starting from the definitions of cosh and sinh in terms of exponentials, prove that
$$\sinh 2 x = 2 \sinh x \cosh x$$
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Using the substitution \(\mathrm { u } = \sinh \mathrm { x }\), find \(\int \sinh ^ { 2 } 2 x \cosh x \mathrm {~d} x\).
Find the particular solution of the differential equation
$$\frac { d y } { d x } + y \tanh x = \sinh ^ { 2 } 2 x$$
given that \(y = 4\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).