= Solution
For a differentiable scalar function and a differentiable curve, the multivariable <chain rule> is
$$
\boxed{\frac{d}{dt}f(x(t),y(t))=f_x\frac{dx}{dt}+f_y\frac{dy}{dt}.}
$$
Let $L=x\partial_x-2y\partial_y$. Choose $u=\alpha(x)y$ to be an invariant of the <method of characteristics>. Then $Lu=y\{x\alpha'(x)-2\alpha(x)\}$, so $x\alpha'=2\alpha$ and a convenient choice is \b[$\alpha(x)=x^2$]. For $v=y/x$, the <chain rule> gives $Lv=-3v$. Thus, writing the unknown in the new coordinates,
$$
Lf=(Lu)f_u+(Lv)f_v=-3v f_v=6f.
$$
Integrating with $u$ fixed yields
$$
\boxed{f=v^{-2}F(u)=\frac{x^2}{y^2}F(x^2y),\qquad xy\ne0,}
$$
where $F$ 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 $F(u)=u^2G(u)$ gives the simpler chart expression
$$
\boxed{f(x,y)=x^6G(x^2y)\quad(x\ne0).}
$$
It is the general solution on either component $x>0$ or $x<0$, including $y=0$ when $G$ is differentiable there. Indeed $(x,u=x^2y)$ is a nonsingular <coordinate chart> for $x\ne0$, and the <weighted Euler first-order equation> reduces to $x\partial_xf|_u=6f$.
For a patch crossing $x=0$ with $y\ne0$, 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.
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