Find the particular solution of the following differential equation: $c o s y d x + (1 + 2 e^{- x}) s i n y d y = 0; y (0) = \frac{π}{4}$ .

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Solution

Here $c o s y d x + (1 + 2 e^{- x}) s i n y d y = 0 \Rightarrow \int \frac{d x}{1 + 2 e^{- x}} = - \int \frac{s i n y}{c o s y} d y \Rightarrow \int \frac{e^{x} d x}{e^{x} + 2} = - \int \frac{s i n y}{c o s y} d y$
In first integral, put $e^{x} + 2 = t \Rightarrow e^{x} d x = d t$
Also, in second integral, put $c o s y = u \Rightarrow - s i n y d y = d u$
$∴ \int \frac{d t}{t} = \int \frac{d u}{u} \Rightarrow l o g | t | = l o g | u | + l o g C$
$\Rightarrow l o g | e^{x} + 2 | = l o g | C c o s y | \Rightarrow e^{x} + 2 = C c o s y$
As $y (0) = \frac{π}{4}, s o e^{0} + 2 = C c o s \frac{π}{4} \Rightarrow C = 3 \sqrt{2}$
Hence the required particular solution is: $e^{x} + 2 = 3 \sqrt{2} c o s y .$

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