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

Starting with the differential equation:
\[(e^y + 1) \cos x \, dx + e^y \sin x \, dy = 0\]
Rewrite as:
\[\implies d \left( (e^y + 1) \sin x \right) = 0\]
Integrating, we get:
\[(e^y + 1) \sin x = C\]
Since the solution passes through \( \left( \frac{\pi}{2}, 0 \right) \), substitute \( x = \frac{\pi}{2} \) and \( y = 0 \):
\[e^0 + 1 = C \implies C = 2\]
Now, let \( x = \frac{\pi}{6} \):
\[(e^y + 1) \sin \frac{\pi}{6} = 2\]
\[\implies \frac{e^y + 1}{2} = 2\]
\[\implies e^y = 3\]
Thus, \( e^{y \left( \frac{\pi}{6} \right)} = 3 \).