= Quadrant uniqueness for unequal dilation characteristics
{title2=$xz_x+2yz_y=xy,\quad z(s,s)=s^2$}
The <method of characteristics> from $s\neq0$ are $(x,y,z)=(se^t,se^{2t},s^2(2+e^{3t})/3)$. Their projections are half-parabolas $y=x^2/s$ with $x/s>0$. The inverse parameters $s=x^2/y$, $t=\log(y/x)$ exist exactly on the first and third quadrants, giving the unique solution $z=2x^4/(3y^2)+xy/3$ there. The projection Jacobian $se^{3t}$ is nonzero, so every allowed initial point is noncharacteristic.
Back to article page
