Question

(y-z)z_x -(x-z)z_y=(y-x)z

Answer 1

Lagrange’s Partial Differential Equation of Order One

$$(y − z)p - (x-z)q = (y-x)z$$

Lagrange’s auxiliary equations

$$\dfrac{dx}{y-z}=\dfrac{dy}{z-x}=\dfrac{dz}{z(y-x)}$$

$$\dfrac{xdx+ydy-dz}{xy-xz+yz-xy-yz+xz}=\dfrac{d(\dfrac{x^2}{2}+\dfrac{y^2}{2}-z)}{0}$$

$$d(\dfrac{x^2}{2}+\dfrac{y^2}{2}-z)=0$$

Integrating

$$\dfrac{x^2}{2}+\dfrac{y^2}{2}-z=a$$

$$\dfrac{dx+dy-\dfrac{1}{z}dz}{y-z+z-x-y+x}=\dfrac{d(x+y-\ln|z|)}{0}$$

$$d(x+y-\ln|z|))=0$$

Integrating

$$x+y-\ln|z|=b$$

The general solution is

$$\phi(\dfrac{x^2}{2}+\dfrac{y^2}{2}-z, x+y-\ln|z|)=0$$