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Provide solution for RD Sharma maths class 12 chapter Differential Equations exercise 21.7 question 14

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Answer: \sin y=e^{x} \log x+c

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

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

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

         \cos y d y=\frac{x \cdot e^{x} \log x+e^{x}}{x} d x

          Integrating both sides

        \begin{aligned} &\int \cos y d y=\int e^{x} \log x d x+\int \frac{e^{x}}{x} d x \\\\ &\sin y=\log x \int e^{x} d x-\int \frac{1}{x}\left(\int e^{x} d x\right) d x+\int \frac{e^{x}}{x} d x \end{aligned}

        \begin{aligned} &\sin y=e^{x} \log x-\int \frac{e^{x}}{x} d x+\int \frac{e^{x}}{x} d x+c \\\\ &\sin y=e^{x} \log x+c \end{aligned}

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