 Solution

The real logarithm requires $x>0$. The identically zero function is a solution and must be retained before dividing by $y$. For a nonzero solution, divide by $x$ and make the Bernoulli differential equation substitution $u=1/y$. This givesIntegration yields $u/x=-x\ln x+x+C$. Consequentlywith each nonzero branch restricted to an interval where its denominator is nonzero. The right-hand side of the first-order equation is locally Lipschitz in $y$ for $x>0$, so a nonzero solution cannot cross the zero solution at a finite positive $x$.