Question

Solve the differential equation \( y \log y \, dx - x \, dy = 0 \).

Hint:

For separable differential equations, rearrange the terms and integrate both sides.

Answer 1

Rewriting the given equation: \[ y \log y \, dx = x \, dy. \] Rearrange to separate variables: \[ \frac{dy}{dx} = \frac{y \log y}{x}. \] This is a separable differential equation: \[ \frac{dy}{y \log y} = \frac{dx}{x}. \] Integrating both sides: \[ \int \frac{dy}{y \log y} = \int \frac{dx}{x}. \] Using substitution \( u = \log y \), \( du = \frac{dy}{y} \), \[ \int \frac{du}{u} = \log |u| = \log |\log y|. \] So we obtain: \[ \log |\log y| = \log |x| + C. \] Exponentiating both sides, \[ \log y = Cx. \] Taking exponent again, \[ y = e^{Cx}. \]