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

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

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

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

y = c_{1} e^{- 2 x} + c_{2} x e^{- 2 x}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{- 2 x} + c_{2} x e^{- 2 x}

y = c_{1} e^{- 2 x} + c_{2} x e^{- 2 x}

35 y^{^{''''}} + 143 y^{^{''}} + 144 y = 0

y^{^{'''}} - 7 y^{^{''}} + 10 y = 0, y^{^{'}} (0) = 1

7 x^{^{''}} + 3 x^{^{'}} + 7 x = 0, x (0) = - 4, x^{^{'}} (0) = 0

y^{^{'''}} - 21 y^{^{''}} + 128 y^{^{'}} - 240 y = 0

y^{^{''}} - 6 y^{^{'}} + 45 y = 0, y (0) = 4, y^{^{'}} (0) = - 6

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