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}

y^{^{''}} - \frac{1}{2} y^{^{'}} + \frac{1}{16} y = 0

\frac{d ^{2} y}{d t ^{2}} - 4 \frac{d y}{d t} - 5 y = 0

y^{^{''}} - 64 y = 0

3 y^{^{''}} + 11 y^{^{'}} - 7 y = 0

y^{^{''}} - 6 y^{^{'}} - 16 y = 0, y (0) = 3, y^{^{'}} (0) = 4

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}