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

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, $e^{y \left( \frac{\pi}{6} \right)} = 3$.