One parameter solutions to the differential equation $y'(x) = -\frac{x}{1+y}\,e^{-x^2}$.

Question

I've already worked out the solutions to it, 2 explicit functions of x:

y1 = $-1 + \sqrt{c1+1-e^{-x^2}}$

y2 = $-1 + -\sqrt{c1+1-e^{-x^2}}$

What I want to know is, Can you express the general solution as a genuine one-parameter family of solutions (i.e., with the use of $\pm$) by using the sign-function sgn and the absolute value function?

Answer 1

$$2y^\prime(1+y)=-2xe^{-x^2}$$

$$\frac{d}{dx}(y^2 +2y) = \frac{d}{dx} e^{-x^2}$$

$$y^2 + 2y = e^{-x^2} + \text{const.}$$

$$(y+1)^2 = e^{-x^2}+ \text{const.}$$

$$y=\mp\sqrt{e^{-x^2}+c}-1 $$