The order of the differential equation whose general solution is given by $y=(c_1+c_2)cos(x+c_3)-c_4e^{x+c_5}$ where $c_1,c_2,c_3,c_4,c_5$ are arbitrary constants, is:

3

$y=(c_1+c_2)cos(x+c_3)-c_4e^x.e^{c_5}$

Let $c_1+c_2=A;c_3=B;c_4.e^{c_5}=c$

$\left.\begin{matrix}y=A\,cos(x+B)-Ce^x\\y'=-A\,sin(x+B)-Ce^x\\y''=-A\,cos(x+B)-Ce^x\\y'''=A\,sin(x+B)-Ce^x\end{matrix}\right\}\begin{matrix}⇒y+y''=-2ce^x\\y'+y'''=-2ce^x\\⇒y+y''=y'+y'''\end{matrix}$

Hence order is 3.