What is the solution for (D^3-2D^2-D+2)y=0 ?

The solution for (D^3-2D^2-D+2)y=0 is y=c_{1}e^{2x}+c_{2}e^{-x}+c_{3}e^x

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(D^3-2D^2-D+2)y=0

(D^{3} - 2 D^{2} - D + 2) y = 0

y = c_{1} e^{2 x} + c_{2} e^{- x} + c_{3} e^{x}

Solution steps

(D^{3} - 2 D^{2} - D + 2) y = 0

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} e^{- x} + c_{3} e^{x}

y = c_{1} e^{2 x} + c_{2} e^{- x} + c_{3} e^{x}

2 y^{^{'''}} + y^{^{''}} - 8 y^{^{'}} + 5 y = 0

\frac{d ^{2} y}{d t ^{2}} = \frac{d y}{d t}

y^{^{''}} + 5 y^{^{'}} - y = 0

y^{^{'''}} + 9 y^{^{''}} + 20 y^{^{'}} = 0

y^{^{''}} + 8 y^{^{'}} + 4 y = 0, y (0) = 0, y^{^{'}} (0) = 1

What is the solution for (D^3-2D^2-D+2)y=0 ?
The solution for (D^3-2D^2-D+2)y=0 is y=c_{1}e^{2x}+c_{2}e^{-x}+c_{3}e^x