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For a differentiable scalar function and a differentiable curve, the multivariable chain rule isLet $L=x{\partial }_x-2y{\partial }_y$. Choose $u=\alpha (x)y$ to be an invariant of the method of characteristics. Then $Lu=y\{x{\alpha }^{\prime }(x)-2\alpha (x)\}$, so $x{\alpha }^{\prime }=2\alpha$ and a convenient choice is $\alpha (x)=x^2$. For $v=y/x$, the chain rule gives $Lv=-3v$. Thus, writing the unknown in the new coordinates,Integrating with $u$ fixed yieldswhere $F$ is an arbitrary differentiable function on the relevant interval. This gives all local solutions on each patch where the specified change of variables is invertible: its Jacobian determinant is $\partial (u,v)/\partial (x,y)=3y$.

The axes need care because $v$ is undefined at $x=0$ and the Jacobian vanishes at $y=0$. Reparameterizing $F(u)=u^{2}G(u)$ gives the simpler chart expressionIt is the general solution on either component $x>0$ or $x<0$, including $y=0$ when $G$ is differentiable there. Indeed $(x,u=x^{2}y)$ is a nonsingular coordinate chart for $x\ne 0$, and the weighted Euler first-order equation reduces to $x{\partial }_{x}f{|}_u=6f$.

For a patch crossing $x=0$ with $y\ne 0$, the characteristic invariant $w=x\sqrt{|y|}$ instead gives $f=|y{|}^{-3}K(w)$, with an arbitrary differentiable function on each sign component of $y$. These chart descriptions must agree on overlaps. On the axes the original equation gives $f(x,0)=C_{\pm }{x}^6$ and $f(0,y)=D_{\pm }|y{|}^{-3}$ on their separate components; a solution defined smoothly through the origin must have $f(0,0)=0$ and compatible smooth limits, in particular $D_{\pm }=0$. No boundary data select a particular arbitrary function.