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

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

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

linear

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

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

Solution steps

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

Solve linear ODE:

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

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

5 y^{^{''}} - 6 y^{^{'}} + y = 0, y (0) = 5, y^{^{'}} (0) = \frac{1}{5}

x^{^{''}} + 4 x^{^{'}} + 8 x = 0

9 y^{^{''}} + 25 y = 0

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

y^{^{''}} + 13 y^{^{'}} = 0, y (0) = - 9, y^{^{'}} (0) = 13

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