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

The solution for (dy)/(dx)=-(y^2)/(1+y^2)e^{-x},y(0)=1 is y=(-e^x+1+sqrt(-2e^x+5e^{2x)+1})/(2e^x)

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(dy)/(dx)=-(y^2)/(1+y^2)e^{-x},y(0)=1

\frac{d y}{d x} = - \frac{y ^{2}}{1 + y ^{2}} e^{- x}, y (0) = 1

y = \frac{- e ^{x} + 1 + - 2 e ^{x} + 5 e ^{2 x} + 1}{2 e ^{x}}

Solution steps

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

Solve separable ODE:

y = \frac{- e ^{x} + 1 + - 2 e ^{x} + 5 e ^{2 x} + 1}{2 e ^{x}}

y = \frac{- e ^{x} + 1 + - 2 e ^{x} + 5 e ^{2 x} + 1}{2 e ^{x}}

\int (x + 1) (x + 3)^{2} d x

\frac{d}{d x} (\frac{32}{x})

- x^{2} + 9, 1.25 x + 3

y^{^{'}} - y = 9 t e^{2 t}, y (0) = 1

x \to 0 - lim (\frac{1}{x \cdot e ^{\frac{1}{x}}})

What is the solution for (dy)/(dx)=-(y^2)/(1+y^2)e^{-x},y(0)=1 ?
The solution for (dy)/(dx)=-(y^2)/(1+y^2)e^{-x},y(0)=1 is y=(-e^x+1+sqrt(-2e^x+5e^{2x)+1})/(2e^x)