What is the solution for (d^8y)/(dx^8)-8(d^4y)/(dx^4)+16y=0 ?

The solution for (d^8y)/(dx^8)-8(d^4y)/(dx^4)+16y=0 is

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(d^8y)/(dx^8)-8(d^4y)/(dx^4)+16y=0

\frac{d ^{8} y}{d x ^{8}} - 8 \frac{d ^{4} y}{d x ^{4}} + 16 y = 0

y = c_{1} e^{2 x} + c_{2} x e^{2 x} + c_{3} e^{- 2 x} + c_{4} x e^{- 2 x} + c_{5} cos (2 x) + c_{6} sin (2 x) + x (c_{7} cos (2 x) + c_{8} sin (2 x))

Solution steps

\frac{d ^{8} y}{d x ^{8}} - 8 \frac{d ^{4} y}{d x ^{4}} + 16 y = 0

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} x e^{2 x} + c_{3} e^{- 2 x} + c_{4} x e^{- 2 x} + c_{5} cos (2 x) + c_{6} sin (2 x) + x (c_{7} cos (2 x) + c_{8} sin (2 x))

y = c_{1} e^{2 x} + c_{2} x e^{2 x} + c_{3} e^{- 2 x} + c_{4} x e^{- 2 x} + c_{5} cos (2 x) + c_{6} sin (2 x) + x (c_{7} cos (2 x) + c_{8} sin (2 x))

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

y^{^{''}} + 4 y - 3 y^{^{'}} = 0

- 100 y + 65 y^{^{'}} - 14 y^{^{''}} + y^{^{'''}} = 0

x^{^{''}} + 4 x^{^{'}} = - 2 x^{^{'}} - 8 x + 3 x

12 y^{^{''}} - 5 y - 2 y = 0

What is the solution for (d^8y)/(dx^8)-8(d^4y)/(dx^4)+16y=0 ?
The solution for (d^8y)/(dx^8)-8(d^4y)/(dx^4)+16y=0 is