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Question
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Solution
We have,
\[\frac{dy}{dx} = e^{x + y} + e^{- x + y} \]
\[ \Rightarrow \frac{dy}{dx} = e^y \left( e^x + e^{- x} \right)\]
\[ \Rightarrow e^{- y} dy = \left( e^x + e^{- x} \right) dx\]
Integrating both sides, we get
\[\int e^{- y} dy = \int\left( e^x + e^{- x} \right) dx\]
\[ \Rightarrow - e^{- y} = e^x - e^{- x} + C\]
\[ \Rightarrow e^{- x} - e^{- y} = e^x + C\]
\[\text{ Hence, } e^{- x} - e^{- y} = e^x + C\text{ is the required solution.} \]
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