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

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

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y^'+x*e^{-x^{2/2}}=0

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

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

Solution steps

y^{'} (x) + x \cdot e^{- x^{\frac{2}{2}}} = 0

Isolate

y^{'} (x) : y^{'} (x) = - e^{- x} x

f^{^{'}} (x) = g (x)

then

f (x) = \int g (x) d x

y = \int - e^{- x} x d x

\int - e^{- x} x d x = e^{- x} x + e^{- x} + c_{1}

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

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