The differential equation whose general solution is given by `y=c_1cos(x+c_2)-c_3e^((-x+c4))+(c_5sinx),` where `c_1,c_2,c_3,c_4,c_5` are arbitrary constants, is

Explanation

`y=c_(1)cos(x+c_(2))-(c_(3)e^(-x+c_(4)))+(c_(5)sinx)`
`=c_(1)(cosxcosc_(2))cosx-(c_(1)sinc_(2)-c_(5))sinx-(c_(3)e^(c^(4))e^(-x))+(c_(5)sinx)`
`=(c_(1)cos c_(2)) cosx-(c_(1)sinc_(2)-c_(5))sinx-(c_(3)e^(c^(4)))e^(-x)`
or `y=lcosx + msin x-ne^(-x)`............(1)
where l,m,n are arbitary constants
`therefore (dy)/(dx)=-lsinx+mcosx+me^(x)`.............(2)
or `(d^(2)y)/(dx^(2))=-lcosx-msinx-ne^(-x)`.............(3)
or `(d^(3)y)(dx^(3))=lsinx-mcosx+ne^(-x)`............(4)
From equations (1) + (3) , `(d^(2)y)/(dx^(2))+y=-2ne^(-x)`............(5)
From equations (2) +(4), `(d^(3)y)/(dx^(3))+(dy)/(dx)=2ne^(-x)`.............(6)
From equations (5) + (6), `(d^(3)y)/(dx^(3))+(dy)/(dx)+y=0`