Solve the system of partial differential equations (∂ z)/(∂…
Solve the system of partial differential equations \[ \frac{\partial z}{\partial x} = y - z, \qquad \frac{\partial z}{\partial y} = xz. \]
Explanation of Method
Before attempting to integrate, check the integrability (compatibility) condition $\partial_y(z_x)=\partial_x(z_y)$. If it forces a constraint that a genuine solution of the system cannot also satisfy, the system has no solution on any open domain.
Step 1 · Compatibility condition
Requiring $\partial_y(y-z)=\partial_x(xz)$: \[ 1-z_y = z+xz_x. \] Substituting the equations themselves ($z_x=y-z$, $z_y=xz$): \[ 1-xz = z+x(y-z) = z+xy-xz \;\Longrightarrow\; 1 = z+xy. \] So any solution of the system would have to satisfy $z=1-xy$ identically.
Step 2 · Test $z=1-xy$ against the original equations
If $z=1-xy$: $z_x=-y$, while $y-z=y-(1-xy)=xy+y-1$. Equality $-y=xy+y-1$ would require $1=xy+2y=y(x+2)$ for all $(x,y)$, which is false. So $z=1-xy$ (the only candidate forced by compatibility) does not actually solve the first equation.
Step 3 · Conclusion
Since the compatibility condition forces a specific function $z=1-xy$ that fails to satisfy the original equations, no function $z(x,y)$ can satisfy both PDEs simultaneously on any open set.
Result
Conclusion: