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}))

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

y^{^{''''}} - 3 y^{^{'''}} - 2 y^{^{''}} + 2 y^{^{'}} + 12 y = 0

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

y^{^{''}} - 6 y^{^{'}} + 9 y = 0, y (0) = 5, y^{^{'}} (0) = 12

25 y + 8 y^{^{'}} + y^{^{''}} = 0

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))