Originally Posted by

**bugatti79** All,

I wish to check that $\displaystyle u(x,2x)=0$ for the following problem

$\displaystyle (y-u)\frac{\partial u}{\partial x}+(u-x)\frac{\partial u}{\partial y}=(x-y)$,

to which the general solution is defined implicitly by

$\displaystyle x+y+u=3 \sqrt { 2( \frac {(x-y)^2}{2}-\frac{2xu +2yu-u^2}{2})$

Using IC's , I reduce it to

$\displaystyle 3x+u=3 \sqrt{u^2-6xu-x^2}$

reducing further, I get

$\displaystyle 3x+u=3 \sqrt {3x+- \sqrt {2} 2x}$

I am having difficulty showing u=0 from here. Mathematica shows 2 solutions u=0 and I think the other is 15x/6.

Thanks