Find the perpendicular solution of the following differential equation
$cos y d x + (1 + {2 e}^{- x}) sin y d y = 0; y (0) = \frac{π}{4}$

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Solution

$cos y d x + (1 + 2 e^{- x}) sin y d y = 0$
$cos y d x = - (1 + 2 e^{- x}) sin y d y$
$\frac{d x}{(1 + 2 e^{- x})} = - \frac{sin y}{cos y} d y$
$\frac{d x}{(1 + 2 e^{- x})} = - tan y d y$
$\frac{d x}{(1 + \frac{2}{e^{x}})} = - tan y d y$
$\frac{d x}{(\frac{e^{x} + 2}{e^{x}})} = - tan y d y$

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