<s>[INST] General Solution for a PDE I would like to know if someone could provide me the solutions for the next PDE:
\begin{align*}
f_{x}(x,y) - f_{y}(x,y) = \frac{y-x}{x^{2}+y^{2}}f(x,y)
\end{align*}
I empirically found the following particular solution given by $f(x,y) = \displaystyle\frac{x+y}{\sqrt{x^{2}+y^{2}}}$. Any contribution is appreciated. Thank you in advance. 
 [/INST] One can use method of characteristics $\gamma = \left(x(t),y(t)\right)$:
$$f_x + f_y = \dfrac{y-x}{x^2+y^2}\,f
$$
$$
x' = y'=1, \qquad \frac{df}{dt} = \dfrac{y-x}{x^2+y^2}\,f
$$
Solving system of ODEs above get
$$
x = t+a,\quad y = t+b = x + c, \implies c = y-x \quad - \quad \operatorname{const}\\
 f' = \frac{f}{2t^2 + 2(a+b)t + a^2 + b^2 }
\\
\ln f = \int \frac{dt}{2t^2 + 2(a+b)t + a^2 + b^2 }
\implies f = \exp \int \frac{dt}{2t^2 + 2(a+b)t + a^2 + b^2 }
$$
Hope you can proceed from here with computing the integral.
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