Word Count : 620

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दी गई समीकरण है: [ (x^2y - 2xy^2)dx - (x^3 - 3x^2y)dy = 0 ] सबसे पहले इसे इस रूप में लिखते हैं: [ M,dx + N,dy = 0 ] जहाँ [ M = x^2y - 2xy^2 = xy(x - 2y), \quad N = -(x^3 - 3x^2y) = -x^2(x - 3y) ] चलिए देखें कि यह exact (सटीक) है या नहीं। Exact होने की शर्त है: [ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} ] अब (\partial M / \partial y) निकालते हैं: [ M = x^2y - 2xy^2 ] [ \frac{\partial M}{\partial y} = x^2 - 4xy ] और (\partial N / \partial x): [ N = - (x^3 - 3x^2y) = -x^3 + 3x^2y ] [ \frac{\partial N}{\partial x} = -3x^2 + ____ _______ _________ ________ ______ __________.
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