Word Count : 497

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We are asked to verify the integrability of the differential equation and find its integral. Let’s carefully solve it manually, step by step. The given differential equation is: $$ z(x^2 - y^2 - z^2)\,dx + (x + z)xz\,dy + x(z^2 - x^2 - xy)\,dz = 0 $$ Let: $$ M = z(x^2 - y^2 - z^2), \quad N = (x + z)xz, \quad P = x(z^2 - x^2 - xy) $$ where $M$ is the coefficient of $dx$, $N$ of $dy$, $P$ _________ _____ ______ ___ ______ _______ _______ ________.
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