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Explain solution RD Sharma class 12 chapter Differential Equations exercise 21.7 question 20 maths

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Answer: y \sin y=x^{2} \log x+c

Hint: You must know about the rules of solving differential equation and integration

Given: \frac{d y}{d x}=\frac{x(2 \log x+1)}{\sin y+y \cos y}

Solution: \frac{d y}{d x}=\frac{x(2 \log x+1)}{\sin y+y \cos y}

        (\sin y+y \cos y) d y=x(2 \log x+1) d x

          Integrating both sides

        \int \sin y d y+\int y \cos y d y=2 \int x \log x d x+\int x d x

        \begin{aligned} &-\cos y+y \int \cos y d y-\int\left[\frac{d}{d y} y \cdot \int \cos y d y\right] d y=2\left\{\log x \int x-\int\left[\frac{d}{d x} \log x \cdot \int x d x\right] d x\right\} \\\\ &-\cos y+[y \sin y+\cos y]=2\left[\frac{x^{2}}{2} \log x-\frac{1}{2} \int x d x\right]+\frac{x^{2}}{2} \\\\ &y \sin y=x^{2} \log x+c \end{aligned}

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