JEE Mains · Maths · STD 12 - 9. differential equations
The solution curve of the differential equation \(y \frac{d x}{d y}=x\left(\log _e x-\log _e y+1\right), x>0, y>0 \text { passing }\) through the point\((\mathrm{e}, 1)\) is
- A \(\left|\log _e \frac{y}{x}\right|=x\)
- B \(\left|\log _e \frac{y}{x}\right|=y^2\)
- C \(\left|\log _e \frac{x}{y}\right|=y\)
- D \(2\left|\log _e \frac{x}{y}\right|=y+1\)
Answer & Solution
Correct Answer
(C) \(\left|\log _e \frac{x}{y}\right|=y\)
Step-by-step Solution
Detailed explanation
\(\frac{d x}{d y}=\frac{x}{y}\left(\ln \left(\frac{x}{y}\right)+1\right)\) \(\text { Let } \frac{x}{y}=t \Rightarrow x=t y\) \(\frac{d x}{d y}=t+y \frac{d t}{d y}\) \(t+y \frac{d t}{d y}=t(\ln (t)+1)\)…
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