What is the solution for x*(dy)/(dx)-x^2+x*e^{-x}=0 ?

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

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x(dy)/(dx)-x^2+xe^{-x}=0

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

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

Solution steps

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

Substitute

\frac{d y}{d x}

with

y^{'} (x)

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

Isolate

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

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

then

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

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

\int x - e^{- x} d x = \frac{x ^{2}}{2} + e^{- x} + c_{1}

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

\frac{d}{d x} (\frac{x ^{5}}{5})

\frac{d}{d x} (\frac{6}{x + 1})

\frac{d}{d x} (cosh (x) - (2 e^{(- 0.3 x)}))

\frac{d P}{d t} = 0.1 P (1 - \frac{P}{830}) - \frac{1720}{83}

\frac{d}{d y} (e^{x} \cdot sin (y))

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