Hi! I ran into this PDE working on a question in cake cutting. Here it is:


$x\partial\_1f(x,y)-(1-x)\partial\_2f(y,x)=0$


for all $(x,y) \in [0,1]\times[0,1]$.


Thanks!



Help with what is most likely an easy PDE
The updated PDE is simpler than the original one. The same argument I gave in the comments still applies, but now you don't need to solve a wave equation, just integrate along one of the coordinates.


Namely, give $f(x,y)$ any sufficiently regular value on $(x,y)\in[0,1]\times[0,1]$ with $y\ge x$. On the other half of the domain $(x,y)\in[0,1]\times[0,1]$ with $x\ge y$, define $g(x,y) = \frac{(1-x)}{x}\partial\_2 f(y,x)$. On that domain your PDE reduces to
$$ \partial\_1 f(x,y) = g(x,y) $$
$$ f(x,y) = f(y,y) + \int\_y^x g(x',y) dx' $$


