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^{^{''}} + 16 y^{^{'}} + 63 y = 0

x^{^{''}} + 81 x = 0

\frac{d ^{2} y}{d x ^{2}} - 10 \frac{d y}{d x} + 26 y = 0

y^{^{'}} = 8 y^{^{''}}

y^{^{''}} - 4 y^{^{'}} + 3 y = 0, y (0) = - 1, y^{^{'}} (0) = 3

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))