Chapter 8 - Mathematical Modeling With Differential Equations - 8.2 Separation Of Variables - Exercises Set 8.2 - Page 575: 3

$$y=c_1e^{-\sqrt{1+x^2} }-1$$

Given $$\frac{\sqrt{1+x^2}}{1+y}\frac{dy}{dx}=-x $$ Separate variables \begin{align*} \int \frac{dy}{1+y }&= \int \frac{-x}{\sqrt{1+x^2}} dx \\ \ln( y+1)&=-\sqrt{1+x^2}+c\\ y&= e^{[c-\sqrt{1+x^2}]}-1\\ &=c_1e^{-\sqrt{1+x^2} }-1 \end{align*}