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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?

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up vote 1 down vote accepted

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

$f_{xy} = xf+(x+y)xf_x$


$f_{yx} = yf+(x+y)yf_y$,

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 $f \equiv 0$.

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thanks, thats a really fruitful way of looking at it. I knew I had to ask a mathematician :) – Tony Nov 19 '12 at 10:28

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