 Solution

Use separation of variables with $u=X(x)Y(y)$ on a region where the factors do not vanish. The equation becomes $X^{\prime }Y-xX{Y}^{\prime }=0$, soThe separated equations are $X^{\prime }=\lambda xX$ and $Y^{\prime }=\lambda Y$, giving $u=C\exp [\lambda (y+x^2/2)]$. Choosing $C=\lambda =1$ gives the nonconstant real solutionAlthough the division used $x\ne 0$, the resulting solution extends smoothly across $x=0$ and directly satisfies $u_x=x{u}_y$ everywhere.