Explanation

(b)
Given, $y = \left( c_{1} + c_{2} \right)\cos{(x + c_{3})} - c_{4}e^{x + c_{5}}$
$\Rightarrow$ $y = \left( c_{1}\cos c_{3} + c_{2}\cos c_{3} \right)\cos x$
${- (c_{1}sin\ c_{3} + c_{2}sin\ c_{3})sin}x - c_{4}e^{c_{5}}e^{x}$
$\Rightarrow$ $y = A\cos x - B\sin x + Ce^{x}$
Where, $A = c_{1}\cos c_{3} + c_{2}\cos c_{3}$
$B = c_{1}\sin c_{3} + c_{2}\sin c_{3}$
And $C = - c_{4}e^{c_{5}}$
Which is an equation containing three arbitrary constant. Hence, the order of the differential equation is 3.