Question

If the solution $y(x)$ of the given differential equation \[(e^y + 1) \cos x \, dx + e^y \sin x \, dy = 0\]passes through the point $\left(\frac{\pi}{2}, 0\right)$, then the value of $e^{y\left(\frac{\pi}{6}\right)}$ is equal to ________.

Answer 1

Correct Answer: 3

Given the differential equation:
\[(e^y + 1) \cos x \, dx + e^y \sin x \, dy = 0\]
This can be rewritten as:
\[\implies d \left( (e^y + 1) \sin x \right) = 0\]
Integration yields:
\[(e^y + 1) \sin x = C\]
The solution passes through \( \left( \frac{\pi}{2}, 0 \right) \). Substituting \( x = \frac{\pi}{2} \) and \( y = 0 \):
\[e^0 + 1 = C \implies C = 2\]
For \( x = \frac{\pi}{6} \):
\[(e^y + 1) \sin \frac{\pi}{6} = 2\]
\[\implies \frac{e^y + 1}{2} = 2\]
\[\implies e^y = 3\]
Therefore, \( e^{y \left( \frac{\pi}{6} \right)} = 3 \).