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Please solve RD Sharma class 12 chapter Differential Equations exercise 21.7 question 13 maths textbook solution

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Answer: x c=\cos y

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

Given: x \frac{d y}{d x}+\cot y=0

Solution: x \frac{d y}{d x}+\cot y=0

        \begin{aligned} &x \frac{d y}{d x}=-\cot y \\\\ &d y=-\cot y \frac{d x}{x} \\\\ &\frac{d y}{-\cot y}=\frac{d x}{x} \end{aligned}

          Integrating both sides

        \begin{aligned} &-\tan y d y=\frac{d x}{x} \\\\ &-\int \tan y d y=\int \frac{d x}{x} \end{aligned}

        \begin{aligned} &-[-\log |\cos y|]=\log |x|+\log C \\\\ &\log |\cos y|=\log |x|+\log C \\\\ &\cos y=x C \end{aligned}

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