The order of the differential equation whose solution is `ae^(x) + be^(2x) + ce^(3x) + d = 0` is

Options

A. `y'''-6y''+11y'-6y=0`
B. `y'''+6y''+11y'+6y=0`
C. `y'''6y''-11y'+6y=0`
D. `y'''-6y''-11y'+6y=0`

Answer

c

Explanation

Given , ` y = Ae^(x) + Be^(2x) +Ce^(3x)" "`…(i)
On differentiating w.r.t. 'x' we get
` rArr y' = Ae^(x) + 2 Be^(2x) + 3Ce^(3x)`
From Eq.(i) , `Ae^(x) = y - Be^(2x) - Ce^(3x)`
` :. Y' = y - Be^(-2x) - Ce^(3x) + 2Be^(2x) + 3Ce^(3x)`
` rArr y ' = y + Be^(2x) + 2Ce^(3x)" "` ... (ii)
Again , on differentiating w.r.t.'x' we get
Again , on differentiating w.r.t.'x' ,we get
` rArr y '' = y' + 2Be^(2x) +6 Ce^(3x)`
From Eq. (ii)
` Be^(2x) = y' -y - 2Ce^(3x)`
` :. y '' = y' +2y' -2y - 4Ce^(3x) + 6 Ce^(3x)`
` rArr y'' = 3y' - 2y + 2 Ce^(3x) " "` ... (iii) Again , on differentiating w.r.t 'x' , we get
` y ''' = 3y'' - 2y' +6 Ce^(3x)`
From Eq. (iii) ,
` 2Ce^(3x) = y'' - 3y' + 2y`
` :. y''' = 3y'' - 3y' + 3( y'' - 3y ' + 2y)`
` rArr y ''' - 6y '' + 11 y' - 6y = 0`