What is the solution for y^'+(2x)/(1+x^2)y=(e^{-x})/(1+x^2) ?

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

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y^'+(2x)/(1+x^2)y=(e^{-x})/(1+x^2)

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

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

Solution steps

y^{'} (x) + \frac{2 x}{1 + x ^{2}} y = \frac{e ^{- x}}{1 + x ^{2}}

Solve linear ODE:

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

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

x \to - 1 lim (\frac{- 1 ^{2} + 1}{- 1})

x \to \infty lim (x^{- 1})

x^{\frac{4}{5}}

(3 x y^{2} + 13 y) + (3 x^{2} y + 13 x) y^{^{'}} = 0

\int x^{2} - 1 d x

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