Differential Eqn. Elimination of arbitrary constant and finding the general solution

Problem 1
$y =C_1 e^{2x} + C_2 e^x + C_3$   ←   eq (1)

There are 3 arbitrary constants C₁, C₂, and C₃, hence, we can differentiate eq (1) up to three times.
$y' = 2C_1 e^{2x} + C_2 e^x$   ←   eq (2)

$y'' = 4C_1 e^{2x} + C_2 e^x$   ←   eq (3)

$y''' = 8C_1 e^{2x} + C_2 e^x$   ←   eq (4)
 

eq (3) - eq(2)
$y'' - y' = 2C_1 e^{2x}$   ←   eq (5)

eq (4) - eq(3)
$y''' - y'' = 4C_1 e^{2x}$   ←   eq (6)

eq (6) - 2×eq (5)
$(y''' - y'') - 2(y'' - y') = 0$

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

$y''' - 3y'' + 2y' = 0$       answer