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Explain solution RD Sharma class 12 chapter Differential Equations exercise 21.7 question 8 maths

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Answer: -e^{-y}=e^{x}+\frac{x^{3}}{3}+3+c

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

Given: \frac{d y}{d x}=e^{x+y}+x^{2} e^{y}

Solution: \frac{d y}{d x}=e^{x+y}+x^{2} e^{y}

        \begin{aligned} &\frac{d y}{d x}=e^{x} \cdot e^{y}+x^{2} e^{y} \\\\ &d y=e^{y}\left[e^{x}+x^{2}\right] d x \\\\ &\frac{d y}{e^{y}}=\left[e^{x}+x^{2}\right] d x \end{aligned}

        Integrating both sides

        \begin{array}{r} \int e^{-y} d y=\int e^{x} d x+\int x^{2} d x \\\\ -e^{-y}=e^{x}+\frac{x^{3}}{3}+3+c \end{array}

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