Word Count : 581

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दी गई अवकल समीकरण है: [ (x^2y - 2xy^2)dx - (x^3 - 3x^2y)dy = 0 ] इसे इस रूप में लिखें: [ M(x,y)dx + N(x,y)dy = 0 ] जहाँ, [ M(x,y) = x^2y - 2xy^2, \quad N(x,y) = -(x^3 - 3x^2y) = -x^3 + 3x^2y ] अब देखें कि यह अवकल समीकरण सटीक (exact) है या नहीं। सटीकता की जाँच के लिए: [ \frac{\partial M}{\partial y} \stackrel{?}{=} \frac{\partial N}{\partial x} ] [ \frac{\partial M}{\partial y} = \frac{\partial}{\partial y}(x^2y - 2xy^2) = x^2 - 4xy ] [ \frac{\partial N}{\partial x} = \frac{\partial}{\partial x}(-x^3 + 3x^2y) = -3x^2 ___ __________ ____ ___ _________ __________ _________ _______ _______ ______.
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