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Linear Differential Equations of First Order

The solution ...

Question

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

\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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x y = log c y

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Solution

The correct option is B $- \frac{1}{x y} + log y = c$
$y d x + (x + x^{2} y) d y = 0 \Rightarrow y d x + x d y = x^{2} y d y \Rightarrow \frac{1}{{(x y)}^{2}} d (x y) = - \frac{d y}{y}$ $[y d x + x d y = d (x y)]$
Integrating both sides, we get
$- \frac{1}{x y} = - log y + c \Rightarrow - \frac{1}{x y} + log y = c$

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