What is the solution for (D^3-1)y=0 ?

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

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(D^3-1)y=0

(D^{3} - 1) y = 0

y = c_{1} e^{x} + e^{- \frac{x}{2}} (c_{2} cos (\frac{3 x}{2}) + c_{3} sin (\frac{3 x}{2}))

Solution steps

(D^{3} - 1) y = 0

Solve linear ODE:

y = c_{1} e^{x} + e^{- \frac{x}{2}} (c_{2} cos (\frac{3 x}{2}) + c_{3} sin (\frac{3 x}{2}))

y = c_{1} e^{x} + e^{- \frac{x}{2}} (c_{2} cos (\frac{3 x}{2}) + c_{3} sin (\frac{3 x}{2}))

28 y^{^{''}} + 17 y^{^{'}} - 3 y = 0

x^{^{''}} + 6 x^{^{'}} + 25 x = 0, x (0) = \frac{1}{2}, x^{^{'}} (0) = 0

4 y^{^{''}} - 12 y^{^{'}} + 9 y = 0, y^{1} (0) = e^{\frac{3 x}{2}}

81 y^{^{''}} - 90 y^{^{'}} + 106 y = 0

y^{^{''}} + 4 y^{^{'}} + 13 y = 0, y (0) = 4, y^{^{'}} (0) = - 7

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