What is the solution for (D^4+6D^3+9D^2)y=0 ?

The solution for (D^4+6D^3+9D^2)y=0 is y=c_{1}+c_{2}x+c_{3}e^{-3x}+c_{4}xe^{-3x}

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(D^4+6D^3+9D^2)y=0

(D^{4} + 6 D^{3} + 9 D^{2}) y = 0

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

Solution steps

(D^{4} + 6 D^{3} + 9 D^{2}) y = 0

Solve linear ODE:

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

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

\frac{d ^{2} y}{d t ^{2}} - 6 \frac{d y}{d t} + 13 y = 0

y^{^{''}} - \frac{3}{10} y^{^{'}} + \frac{1}{50} y = 0

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

y^{^{'''}} - 18 y^{^{''}} + 95 y^{^{'}} - 150 y = 0

\frac{d ^{2} y}{d x ^{2}} + 11 \frac{d y}{d x} + 28 y = 0

What is the solution for (D^4+6D^3+9D^2)y=0 ?
The solution for (D^4+6D^3+9D^2)y=0 is y=c_{1}+c_{2}x+c_{3}e^{-3x}+c_{4}xe^{-3x}