Find the particular solution of the following differential equation.

$\cos y \, dx + (1 + 2e^{-x}) \sin y \, dy = 0; \quad y(0) = \frac{\pi}{4}$

$e^x + 2 = 3\sqrt{2} \cos y$

The correct answer is Option (2) → $e^x + 2 = 3\sqrt{2} \cos y$ ##

$\cos y \, dx + (1 + 2e^{-x}) \sin y \, dy = 0$

$\Rightarrow \int \frac{dx}{1 + 2e^{-x}} = \int \frac{-\sin y}{\cos y} dy$

$\Rightarrow \int \frac{e^x}{2 + e^x} dx = \int \frac{-\sin y}{\cos y} dy$

$\log (e^x + 2) = \log |\cos y| + \log C$

$\log (e^x + 2) = \log |\cos y \cdot C|$

$\Rightarrow e^x + 2 = C |\cos y|$

$\Rightarrow e^x + 2 = \pm C \cos y \Rightarrow e^x + 2 = k \cos y \quad \dots(i)$

Substituting $x = 0, y = \frac{\pi}{4}$ in $(i)$, we get

$1 + 2 = k \cos \frac{\pi}{4}$

$\Rightarrow k = 3\sqrt{2}$

$∴e^x + 2 = 3\sqrt{2} \cos y$

is the particular solution.