Elimination of Arbitrary Constants

$y = c_1 e^{-3x} + c_2 e^{2x}$       → equation (1)

$y' = -3c_1 e^{-3x} + 2c_2 e^{2x}$       → equation (2)

$y'' = 9c_1 e^{-3x} + 4c_2 e^{2x}$       → equation (3)
 

3 × equation (1) + equation (2)
$3y + y' = 5c_2 e^{2x}$       → equation (4)
 

3 × equation (2) + equation (3)
$3y' + y'' = 10c_2 e^{2x}$       → equation (5)
 

2 × equation (4) - equation (5)
$2(3y + y') - (3y' + y'') = 0$

$6y + 2y' - 3y' - y'' = 0$

$6y - y' - y'' = 0$           answer
 

Note: The methods of elimination vary with the way in which the constants enter the given relation.