What is the solution for (d^2y)/(dx^2)+8(dy)/(dx)+16y=0 ?

The solution for (d^2y)/(dx^2)+8(dy)/(dx)+16y=0 is y=c_{1}e^{-4x}+c_{2}xe^{-4x}

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(d^2y)/(dx^2)+8(dy)/(dx)+16y=0

\frac{d ^{2} y}{d x ^{2}} + 8 \frac{d y}{d x} + 16 y = 0

y = c_{1} e^{- 4 x} + c_{2} x e^{- 4 x}

Solution steps

\frac{d ^{2} y}{d x ^{2}} + 8 \frac{d y}{d x} + 16 y = 0

Solve linear ODE:

y = c_{1} e^{- 4 x} + c_{2} x e^{- 4 x}

y = c_{1} e^{- 4 x} + c_{2} x e^{- 4 x}

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

x^{^{''}} + 2 x^{^{'}} - 3 x = 0

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

2 y^{^{''}} + 3 y^{^{'}} - 2 y = 0, y (0) = - 1, y^{^{'}} (0) = - 4

y^{^{'''}} + 3 y^{^{''}} - 4 y^{^{'}} - 12 y = 0

What is the solution for (d^2y)/(dx^2)+8(dy)/(dx)+16y=0 ?
The solution for (d^2y)/(dx^2)+8(dy)/(dx)+16y=0 is y=c_{1}e^{-4x}+c_{2}xe^{-4x}