 Solution

Set $u=x+y$, so $u^{\prime }=1+y^{\prime }=1+u^2/2$. The new equation is a separable differential equation:Absorbing the constant scaling into $C$, the real solutions areEach 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 $u$.