What is the solution for (d^6y)/(dx^6)-2(d^3y)/(dx^3)+y=0 ?

The solution for (d^6y)/(dx^6)-2(d^3y)/(dx^3)+y=0 is

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(d^6y)/(dx^6)-2(d^3y)/(dx^3)+y=0

\frac{d ^{6} y}{d x ^{6}} - 2 \frac{d ^{3} y}{d x ^{3}} + y = 0

y = c_{1} e^{x} + c_{2} x e^{x} + e^{- \frac{x}{2}} (c_{3} cos (\frac{3 x}{2}) + c_{4} sin (\frac{3 x}{2})) + e^{- \frac{x}{2}} x (c_{5} cos (\frac{3 x}{2}) + c_{6} sin (\frac{3 x}{2}))

Solution steps

\frac{d ^{6} y}{d x ^{6}} - 2 \frac{d ^{3} y}{d x ^{3}} + y = 0

Solve linear ODE:

y = c_{1} e^{x} + c_{2} x e^{x} + e^{- \frac{x}{2}} (c_{3} cos (\frac{3 x}{2}) + c_{4} sin (\frac{3 x}{2})) + e^{- \frac{x}{2}} x (c_{5} cos (\frac{3 x}{2}) + c_{6} sin (\frac{3 x}{2}))

y = c_{1} e^{x} + c_{2} x e^{x} + e^{- \frac{x}{2}} (c_{3} cos (\frac{3 x}{2}) + c_{4} sin (\frac{3 x}{2})) + e^{- \frac{x}{2}} x (c_{5} cos (\frac{3 x}{2}) + c_{6} sin (\frac{3 x}{2}))

y^{^{''}} - 4 y^{^{'}} + 4 y = 0, y (0) = - 4, y^{^{'}} (0) = - 10

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

y^{^{''}} - 2 y^{^{'}} + 5 y = 0, y (0) = - 4, y^{^{'}} (0) = 3

x^{^{''}} + 4 x^{^{'}} - 5 x = 0

9 y^{^{''}} - 42 y^{^{'}} + 49 y = 0

What is the solution for (d^6y)/(dx^6)-2(d^3y)/(dx^3)+y=0 ?
The solution for (d^6y)/(dx^6)-2(d^3y)/(dx^3)+y=0 is