|
0
|
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? |
||
|
|
|
1
|
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 $f \equiv 0$. |
|||
|
