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

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

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(d^4y)/(dx^4)+y=0

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{'''}} - 7 y^{^{''}} + 19 y^{^{'}} - 13 y = 0

y^{^{'''}} - 3 y^{^{''}} - 9 y^{^{'}} + 27 y = 0

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

8 \frac{d ^{2} y}{d x ^{2}} + 3 \frac{d y}{d x} + 9 y = 0

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

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