What is the solution for (3D^3+5D^2-2D)y=0 ?

The solution for (3D^3+5D^2-2D)y=0 is y=c_{1}+c_{2}e^{x/3}+c_{3}e^{-2x}

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(3D^3+5D^2-2D)y=0

(3 D^{3} + 5 D^{2} - 2 D) y = 0

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

Solution steps

(3 D^{3} + 5 D^{2} - 2 D) y = 0

Solve linear ODE:

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

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

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

16 y^{^{''}} - 24 y^{^{'}} + 9 y = 0, y (0) = 3, y^{^{'}} (0) = 5

y^{^{'''}} - 3 y^{^{'}} + 2 y = 0

3 y^{^{''}} - 5 y^{^{'}} - 12 y = 0

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

What is the solution for (3D^3+5D^2-2D)y=0 ?
The solution for (3D^3+5D^2-2D)y=0 is y=c_{1}+c_{2}e^{x/3}+c_{3}e^{-2x}