What is the solution for (D^4-2D^3-3D^2+4D+4)y=0 ?

The solution for (D^4-2D^3-3D^2+4D+4)y=0 is y=c_{1}e^{-x}+c_{2}xe^{-x}+c_{3}e^{2x}+c_{4}xe^{2x}

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(D^4-2D^3-3D^2+4D+4)y=0

(D^{4} - 2 D^{3} - 3 D^{2} + 4 D + 4) y = 0

y = c_{1} e^{- x} + c_{2} x e^{- x} + c_{3} e^{2 x} + c_{4} x e^{2 x}

Solution steps

(D^{4} - 2 D^{3} - 3 D^{2} + 4 D + 4) y = 0

Solve linear ODE:

y = c_{1} e^{- x} + c_{2} x e^{- x} + c_{3} e^{2 x} + c_{4} x e^{2 x}

y = c_{1} e^{- x} + c_{2} x e^{- x} + c_{3} e^{2 x} + c_{4} x e^{2 x}

40 y^{^{''}} + 7 y^{^{'}} - 3 y = 0

y^{^{''}} + k^{2} \cdot y = 0

y^{^{''}} = y

y^{^{''}} - 7 y^{^{'}} - 8 y = 0, y (0) = 1, y^{^{'}} (0) = - 1

y^{^{''''}} - 8 y^{^{'''}} + 18 y^{^{''}} - 27 y = 0

What is the solution for (D^4-2D^3-3D^2+4D+4)y=0 ?
The solution for (D^4-2D^3-3D^2+4D+4)y=0 is y=c_{1}e^{-x}+c_{2}xe^{-x}+c_{3}e^{2x}+c_{4}xe^{2x}