Question

Verify that the total differential equation(1+yz)dx+z(z-x)dy-(1+xy)dz=0 is integrable and hence find its integral.

Answer 1

$P(\frac{\partial Q}{\partial z}-\frac{\partial R}{\partial y})+Q(\frac{\partial R}{\partial x}-\frac{\partial P}{\partial z})+R(\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x})=$

$=(1+yz)(2z-x+x)+z(z-x)(-y-2z+x)-(1+xy)(z+z)=$

$=2yz^2-2xyz-z^2y-2z^3+z^2x+xyz+2z^2x-zx^2\neq0$

The equation is not integrable.