6.17 Solution

On integrating, we have

which equals $\frac{1}{2}e^{x^{2}}+c$. Dividing both sides by $e^{x^{2}}$, we obtain solutions $y={\frac{1}{2}+ce^{-x^{2}}}$ where $c$ is an arbitrary constant.

Let’s verify that this gives a solution of the equation: we have $\frac{dy}{dx}={-2cxe^{-x^{2}},}$ so

as required.