Find the differential equation whose general solution is given by `y=(c_(1)+c_(2))cos(x+c_(3))-c_(4)e^(x+c_(5))`, where `c_(1),c_(2), c_(3), c_(4), c_(5)` are arbitary constants.

Answer

`(d^(3)y)/(dx^(3))-(d^(2)y)/(dx^(2))+(dy)/(dx)-y=0`

Explanation

`y=(c_(1)+c_(2))cosx+c_(3)-c_(4)e^(x+c_(5))`
or `y=Acos(x+B)-Ce^(x)`, where
`A=c_(1)+c_(2),B=c_(3)` and `C=c_(4)e^(c_(5))`
`rArr (dy)/(dx)=-Asin(x+B)=Ce^(x)`
`rArr (d^(2)y)/(dx^(2))=-Acos(x+B)-Ce^(x)`
`rArr (d^(2)y)/(dx^(2))+y-2Ce^(x)`
`rArr (d^(2)y)/(dx^(2))+y=-2Ce^(x)`
`rArr (d^(3)y)/(dx^(3))-(d^(2)y)/(dx^(2))+(dy)/(dx)-y=0`