Solution of the differential equation $y\log_e y dx - x dy = 0$ is (Where c is an arbitrary constant)

The differential equation is $y\log_e y \, dx - x \, dy = 0$

Rearranging:

$y\log_e y \, dx = x \, dy$

$\frac{dx}{x} = \frac{dy}{y\log_e y}$

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Integrating both sides:

$\int \frac{dx}{x} = \int \frac{dy}{y\log_e y}$

The left side gives:

$\int \frac{dx}{x} = \log_e |x| + c_1$

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For the right side, let $u = \log_e y$

Then $du = \frac{dy}{y}$

$\int \frac{dy}{y\log_e y} = \int \frac{1}{u} \, du$

$= \log_e |u| + c_2$

$= \log_e |\log_e y| + c_2$

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Combining the results:

$\log_e |x| = \log_e |\log_e y| + C$

where $C = c_2 - c_1$

$\log_e |x| - \log_e |\log_e y| = C$

$\log_e \left(\frac{|x|}{|\log_e y|}\right) = C$

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Taking exponential of both sides:

$\frac{|x|}{|\log_e y|} = e^C$

Let $e^C = |c|$ where $c$ is an arbitrary constant:

$\frac{|x|}{|\log_e y|} = |c|$

$|x| = |c\log_e y|$

Therefore, the solution is $|x| = |c\log_e y|$