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Need solution for RD Sharma maths class 12 chapter Differential Equations exercise 21.7 question 35

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Answer: e^{-x}-e^{-y}-e^{x}=c

Hint: Separate the terms of x and y and then integrate them.

Given: \frac{d y}{d x}=e^{x+y}+e^{-x+y}

Solution: \frac{d y}{d x}=e^{x+y}+e^{-x+y}

        \begin{aligned} &\frac{d y}{d x}=e^{x} e^{y}+e^{-x} e^{y} \\\\ &\frac{d y}{d x}=e^{y}\left[e^{x}+e^{-x}\right] \\\\ &\frac{d y}{e^{y}}=\left[e^{x}+e^{-x}\right] d x \end{aligned}

          Integrating both sides

        \begin{aligned} &\int e^{-y} d y=\int e^{x} d x+\int e^{-x} d x \\\\ &-e^{-y}=e^{x}+\left[-e^{-x}\right]+c \\\\ &-e^{-y}=e^{x}-e^{-x}+c \\\\ &e^{-x}-e^{-y}-e^{x}=c \end{aligned}

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