Hello,


I have what appears at first sight to be a simple system of coupled first order pdes in two unknowns which I need to solve simultaenously:


$ \frac{\partial f(x,y)}{\partial x}=(x+y)\*x\*f(x,y) \quad;\quad
 \frac{\partial f(x,y)}{\partial y}=(x+y)\*y\*f(x,y) $


But there doesnt seem to be a solution that I or mathematica can come up with. Does anyone have a suggestion for a solution. Or conversely a proof that it has no solution?



Coupled first order pdes in two unknowns
The only smooth solution on some smooth, open domain in $\mathbb{R}^2$ is the zero solution. 


Consider the following:


If $f$ solving this equation were smooth, then we would require $f\_{xy}=f\_{yx}$, and so computing


$f\_{xy} = xf+(x+y)xf\_x$


and


$f\_{yx} = yf+(x+y)yf\_y$,


and equating these we would need


$xf+(x+y)^2x^2f = yf+(x+y)^2y^2f$,


and at every point for which $f \neq 0$, this would require


$x+(x+y)^2x^2 = y+(x+y)^2y^2$,


which is not true on any open set, and so the smoothness of $f$ implies $f \equiv 0$.


