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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

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

y^{^{''}} + 4 y^{^{'}} + 5 y = 0, y (0) = 1, y^{^{'}} (0) = 1

y^{^{''}} + 4 y = 0, y (0) = - 5, y^{^{'}} (0) = 2

y^{^{''}} + y = 0, y (\frac{π}{4}) = 0, y^{^{'}} (\frac{π}{4}) = - 3

2 y^{^{''}} + 10 y = 0, y (0) = 1, y^{^{'}} (0) = - 2

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