Weighted Euler first-order equation
= Weighted Euler first-order equation
For $x f_x+\beta y f_y=\gamma f$, the curves in the <method of characteristics> are $x=x_0e^s$, $y=y_0e^{\beta s}$ and $f=f_0e^{\gamma s}$. On either component $x\ne0$, the invariant is $y|x|^{-\beta}$, giving $f=|x|^\gamma F(y|x|^{-\beta})$. A coordinate change that includes $y/x$ may be singular on an axis, so smooth continuation there must be checked against the original <partial differential equation>.