Does a PDE of the form $$y \frac{\partial u}{\partial x}+x \frac{\partial u}{\partial y} = c(x,y) \, u +d(x,y)$$ necessarily have a solution near $(x,y)=(0,0)$?

*My attempt:* The method of characteristics reduces this to the simple ODE $u'=c u +d$ along the characteristic curves, i.e. the integral curves of the vector field $$v(x,y,u)=(y,x,c(x,y)u+d(x,y))$$ on $\mathbb{R}^3_{(x,y,u)}$. It's easy to see that these characteristic curves all lie over hyperbolae in the $xy$-plane (as in the figure below). The problem is that the origin is a sort of "degenerate" characteristic curve. Here's my hope: Use the union of the $x$- and $y$-axes as a singular initial hypersurface by assigning $u=0$ for $x=0$ and $y=0$. Then use integrating factors to solve $u'=cu+d$ along all characteristic curves that meet either the $x$- or $y$-axis. This should give a solution everywhere except the four open rays in the set $X=\{y=\pm x\} \setminus \{(0,0)\}$, which do not meet the $x$- or $y$-axis. Then one can extend this solution to $X$ using continuity, but now you'd need to prove smoothness somehow.