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^{^{'''}} - 4 y^{^{''}} + 6 y^{^{'}} - 4 y = 0

y^{^{'''}} - 5 y^{^{''}} + 17 y^{^{'}} - 13 y = 0

y^{^{''}} - 4 y^{^{'}} + 4 y = 0, y (1) = 1, y^{^{'}} (1) = 1

\frac{d ^{2} y}{d x ^{2}} + 2 \frac{d y}{d x} + 5 y = 0

y^{^{''}} + 22 y^{^{'}} + 122 y = 0, y (0) = 1, y^{^{'}} (0) = 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))