Word Count : 481

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We are asked to solve the differential equation: [ (y^2 + yz),dx + (z^2 + zx),dy + (x^2 + xy),dz = 0 ] First, observe the symmetry in the equation. The terms are cyclic in (x, y, z). Let us try a solution of the form: [ x + y + z = k ] where (k) is a constant. Then (dx + dy + dz = 0). Substitute (z = k - x - y) into the equation. 1. Compute each term: [ y^2 + yz = y^2 + y(k - x - y) = y^2 + yk - xy - y^2 = yk - xy ] [ z^2 + zx = (k - x - y)^2 + (k ___ _______ ____ ________ __________ _______ _______ __________ _____ ______ _________.
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