The solution of the differential equation y dx+(x+x^2y)dy=0 is - Mathematics MCQ AIEEE 2004 2004 | ExamTest.live

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The solution of the differential equation y d x + (x + x^{2} y) d y = 0 is

Solution & Step-by-step Explanation

Given y d x + (x + x^{2} y) d y = 0 . Rearrange the terms: y d x + x d y + x^{2} y d y = 0 . (y d x + x d y) + x^{2} y d y = 0 Recall d (x y) = x d y + y d x . d (x y) + x^{2} y d y = 0 Divide by x^{2} y^{2} : \frac{d ( x y )}{( x y ) ^{2}} + \frac{x ^{2} y d y}{x ^{2} y ^{2}} = 0 . \frac{d ( x y )}{( x y ) ^{2}} + \frac{1}{y} d y = 0 . Integrating both sides: \int (x y)^{- 2} d (x y) + \int \frac{1}{y} d y = C - \frac{1}{x y} + ln y = C .

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The solution of the differential equation y d x + (x + x^{2} y) d y = 0 is

- \frac{1}{x y} = C

- \frac{1}{x y} + lo g y = C

\frac{1}{x y} + lo g y = C

lo g y = C x

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