Word Count : 322

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The given 1-form (writing $x^2$ etc. clearly) $$ M\,dx+N\,dy+P\,dz \;=\; z(x^2-y^2-z^2)\,dx \;+\; xz(x+z)\,dy \;+\; x\big(z^2-x^2-xy\big)\,dz $$ is not exact as written. Direct partial-derivative checks give $$ \frac{\partial M}{\partial y}=-2zy,\qquad \frac{\partial N}{\partial x}=2xz+z^2, $$ so $\partial M/\partial y\neq\partial N/\partial x$; similarly the other pairwise equalities fail. Hence no potential function exists for the form itself. However the three coefficients are homogeneous polynomials of degree $3$. For a homogeneous differential 1-form of common degree an integrating factor can often be taken from __________ _______ ____ _____ ________ __________ _______ __________ ______ __________ ____ _______.
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