Solution of the differential equation $y d x + (x + x^{2} y) d y = 0$ is:

log y = c x

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- \frac{1}{x y} + log y = c

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\frac{1}{x y} + log y = c

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

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Solution

The correct option is D $- \frac{1}{x y} + log y = c$
Given differential equation is
$y d x + (x + x^{2} y) d y = 0$ ....(1)
Comparing with $M d x + N d y = 0$
Here, $M = y; N = 1 + 2 x y$
$\frac{δ M}{δ y} = 1$
$\frac{δ N}{δ x} = 1 + 2 x y$
$\Rightarrow \frac{δ M}{δ y} \neq \frac{δ N}{δ x}$
Hence, the given eqn is not exac.
So, dividing (1) by $x^{2} y^{2}$
$\Rightarrow \frac{y d x + x d y}{x^{2} y^{2}} + \frac{d y}{y} = 0$
$\Rightarrow d (- \frac{1}{x y}) + \frac{d y}{y} = 0$
Integrating , we get
$\Rightarrow \frac{- 1}{x y} + log y = c$

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[EAMCET 2003]

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