For the differential equation in given question find the general solution.

$(e^{x} + e^{- x}) d y - (e^{x} - e^{- x}) d x = 0$

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Solution

Given, $(e^{x} + e^{- x}) d y - (e^{x} - e^{- x}) d x = 0 \Rightarrow (e^{x} + e^{- x}) d y = (e^{x} - e^{- x}) d x = 0$
On separating the variables, we get $d y = (\frac{e^{x} - e^{- x}}{e^{x} + e^{- x}}) d x$
On integrating, we get,
$\int d y = \int (\frac{e^{x} - e^{- x}}{e^{x} + e^{- x}}) d x L e t e^{x} + e^{- x} = t \Rightarrow e^{x} - e^{- x} = \frac{d t}{d x} \Rightarrow d x = \frac{d t}{e^{x} - e^{- x}} ∴ \int d y = \int \frac{e^{x} - e^{- x}}{t} \frac{d t}{e^{x} - e^{- x}} = \int \frac{1}{t} d t \Rightarrow y = l o g | t | + C \Rightarrow y = l o g | e^{x} + e^{- x} | + C$
which is the required general solution.

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