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

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

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

\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}

y^{^{''}} - 4 y^{^{'}} + 4 y = 0, y (0) = 1, y^{^{'}} (0) = 5

y^{^{''''}} + 10 y^{^{''}} + 25 y = 0

36 y^{^{''}} + 12 y^{^{'}} + y = 0

y^{^{'''}} + 3 y^{^{''}} - 54 y = 0, y (0) = e^{3 x}

9 y^{^{'''}} + 11 y^{^{''}} + 4 y^{^{'}} - 14 y = 0

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