Solve the differential equation $(e^{x} + e^{- x}) d y - (e^{x} - e^{- x}) d x = 0$

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Solution

$d y = \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}} d x$
integrating on both sides, we have
$\int d y = \int \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}} d x$
Put $e^{x} + e^{- x} = t ⟹ (e^{x} - e^{- x}) d x = d t$
$∴ \int d y = \int \frac{1}{t} d t$
$∴ y = l o g t + c$
Substituting value of $t$ , we have
$y = l o g (e^{x} + e^{- x}) + c$
This is the required solution.

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