What is the solution for ((d^2y)/(dx^2)+4(dy)/(dx)+20y)^2=0 ?

The solution for ((d^2y)/(dx^2)+4(dy)/(dx)+20y)^2=0 is y=e^{-2x}(c_{1}cos(4x)+c_{2}sin(4x))

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((d^2y)/(dx^2)+4(dy)/(dx)+20y)^2=0

(\frac{d ^{2} y}{d x ^{2}} + 4 \frac{d y}{d x} + 20 y)^{2} = 0

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

Solution steps

(\frac{d ^{2} y}{d x ^{2}} + 4 \frac{d y}{d x} + 20 y)^{2} = 0

Solve linear ODE:

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

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

y^{^{''}} - 9 y^{^{'}} - 22 y = 0

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

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

y^{^{'''}} + 6 y^{^{''}} + 12 y^{^{'}} + 8 y = 0

(D^{3} + 1) y = 0

What is the solution for ((d^2y)/(dx^2)+4(dy)/(dx)+20y)^2=0 ?
The solution for ((d^2y)/(dx^2)+4(dy)/(dx)+20y)^2=0 is y=e^{-2x}(c_{1}cos(4x)+c_{2}sin(4x))