= Solution
Set $u=x+y$, so $u'=1+y'=1+u^2/2$. The new equation is a <separable differential equation>:
$$
\int\frac{du}{1+u^2/2}=x+C,\qquad\sqrt2\arctan\frac u{\sqrt2}=x+C.
$$
Absorbing the constant scaling into $C$, the real solutions are
$$
\boxed{y(x)=\sqrt2\tan\left(\frac x{\sqrt2}+C\right)-x.}
$$
Each solution is taken on a connected interval avoiding its tangent poles. No constant real solution for $u$ was lost by separation, because $1+u^2/2$ is strictly positive for real $u$.
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