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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

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

y^{^{''}} + 4 y^{^{'}} + 29 y = 0, y (0) = 0, y^{^{'}} (0) = 5

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

x^{^{''}} + 2 x^{^{'}} + 1000 x = 0

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

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