The solution curve of the differential equation y \frac{d x}{d y}=x\left(\log _e x-\log _e y+1\right), x>0, y>0 \quad passing through the point (e, 1) is

[JEE (Main) 31 Jan 2024 S I]

(1) \left|\log _e \frac{y}{x}\right|=y^2

(2) \left|\log _e \frac{y}{x}\right|=x

(3) 2\left|\log _e \frac{x}{y}\right|=y+1

(4) \left|\log _e \frac{x}{y}\right|=y

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