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Solution of the differential equation $y \log_e y\, dx-x\, dy = 0$ is (Where c is an arbitrary constant) |
$|y|=|c\log_e(xy)|$ $|x| = |cy|$ $|x| = |c\log_e y|$ $|y| = |c\log_e x|$ |
$|x| = |c\log_e y|$ |
The correct answer is Option (3) → $|x| = |c\log_e y|$ $y \log_e y \, dx - x \, dy = 0$
$y \log_e y \, dx = x \, dy$
$\frac{dx}{x} = \frac{dy}{y \log_e y}$
$\int \frac{1}{x} \, dx = \int \frac{1}{y \log_e y} \, dy$
Left-Hand Side = $\int \frac{1}{x} \, dx = \log_e |x|$
Right-Hand Side :
To evaluate $\int \frac{1}{y \log_e y} \, dy$,
use the substitution method: Let $u = \log_e y$, which gives $du = \frac{1}{y} \, dy$.
$\int \frac{1}{u} \, du = \log_e |u| = \log_e |\log_e y|$
$\log_e |x| = \log_e |\log_e y| + \log_e |c_1|$
$\log_e |x| = \log_e |c_1 \log_e y|$
$|x| = |c_1 \log_e y|$
$|x| = |c \log_e y|$
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