What is the solution for x*y^{''}+2x*y^'+x*y=0 ?

The solution for x*y^{''}+2x*y^'+x*y=0 is y=c_{1}e^{-x}+c_{2}xe^{-x}

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xy^{''}+2xy^'+x*y=0

x \cdot y^{^{''}} + 2 x \cdot y^{^{'}} + x \cdot y = 0

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

Solution steps

x \cdot y^{''} (x) + 2 x \cdot y^{'} (x) + x \cdot 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}

\frac{d ^{2} y}{d x ^{2}} - 3 \frac{d y}{d x} - 4 y = 0

y^{^{''''}} + 2 y^{^{'''}} + y^{^{''}} = 0

y^{^{'''}} - 11 y^{^{''}} + 24 y^{^{'}} + 36 y = 0

y^{^{'''}} - 11 y^{^{''}} + 35 y^{^{'}} - 25 y = 0

y^{^{''''}} - 7 y^{^{'''}} - 49 y^{^{''}} + 343 y^{^{'}} = 0

What is the solution for x*y^{''}+2x*y^'+x*y=0 ?
The solution for x*y^{''}+2x*y^'+x*y=0 is y=c_{1}e^{-x}+c_{2}xe^{-x}