Arbitrary constant

$y = x^2 + c_1 e^{2x} + c_2 e^{3x}$   ←   Equation (1)

$y' = 2x + 2c_1 e^{2x} + 3c_2 e^{3x}$   ←   Equation (2)

$y'' = 2 + 4c_1 e^{2x} + 9c_2 e^{3x}$   ←   Equation (3)
 

Equation (2) - 3 × Equation (1)
$y' - y = (2x - 3x^2) - c_1 e^{2x}$   ←   Equation (4)

Equation (3) - 3 × Equation (2)
$y'' - y' = (2 - 6x) - 2c_1 e^{2x}$   ←   Equation (5)
 

Equation (5) - 2 × Equation (4)
$(y'' - y') - 2(y' - y) = (2 - 6x) - 2(2x - 3x^2)$

$y'' - 3y' + 2y = 2 - 10x + 6x^2$   ←   (answer)