Show that \(( \cosh x + \sinh x ) ^ { \frac { 1 } { 2 } } = \mathrm { e } ^ { \frac { 1 } { 2 } x }\).
Find the particular solution of the differential equation
$$\frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { d y } { d x } + 3 y = 5 ( \cosh x + \sinh x ) ^ { \frac { 1 } { 2 } }$$
given that, when \(x = 0 , y = 1\) and \(\frac { d y } { d x } = \frac { 4 } { 3 }\).