Question

y=c_1e^x+c2e^{2x}+c_3e^{3x},\:y^3-6y^2-6y=0

Answer 1

$y = c_1e^x + c_2e^{2x} +c_3e^{3x}\\ y' = c_1e^x + 2c_2e^{2x} +3c_3e^{3x} \\ y'' = c_1e^x + 4c_2e^{2x} +9c_3e^{3x}\\ y''' = c_1e^x + 8c_2e^{2x} +27c_3e^{3x}$

Substituting into $y'''-6y''-6y$ we have

$c_1e^x + 8c_2e^{2x} +27c_3e^{3x}-6(c_1e^x + 4c_2e^{2x} +9c_3e^{3x}) -6(c_1e^x + c_2e^{2x} +c_3e^{3x})$$= -11c_1e^x -22 c_2e^{2x} -33c_3e^{3x} = -11y' \neq0$

The above equation can only be 0 if y' =0

Otherwise y is not a solution of the given differential equation.