Word Count : 372

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Consider the differential 1-form $$ \omega \;=\; M\,dx+N\,dy+P\,dz $$ with $$ M=z(x^2-y^2-z^2),\qquad N=xz(x+z),\qquad P=x(z^2-x^2-xy). $$ All three functions are homogeneous of the same degree (degree $3$), so a standard approach is to look for an integrating factor built from the Euler combination $xM+yN+zP$. Compute $$ xM+yN+zP = xz(x^2-y^2-z^2)+y\big(x^2z+xz^2\big)+z\big(xz^2-x^3-x^2y\big). $$ Collecting terms gives the cancellation of all cubic symmetric pieces and leaves $$ xM+yN+zP = x y z (z-y). $$ Hence the scalar function $$ ____ __________ ___ _____ ___ ________ __________ ___ ________ __________.
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