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Solving Homogeneous Differential Equations

The solution ...

Question

The solution of the equation
$(x + log y) d y + y d x = 0$ is

x y + y log y = C

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x y + y log y - y = C

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x y + log y - x = C

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None of the above

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Solution

The correct option is A $x y + y log y - y = C$
Given differential equation is $x d y + y d x + log y d y = 0$
$\Rightarrow x d y + y d x = - log y d y$
$\Rightarrow y \frac{d x}{d y} + x = - log y$
$\Rightarrow \frac{d x}{d y} + \frac{x}{y} = - \frac{log y}{y}$
I.F $= e^{\int \frac{1}{y} d y} = y$
Hence, solution is $x . y = - \int y . \frac{log y}{y} d y + C$
$\Rightarrow x y = - (y log y - y) + C$
$\Rightarrow x y + (y log y - y) = C$

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