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Provide solution for RD Sharma maths Class 12 Chapter 21 Differential Equation Exercise 21.10 Question 9 textbook solution.

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Answer : 4 x y=2 x^{2} \log |x|-x^{2}+C

Hint : To solve this equation we will use differentiation method.

Give : x \frac{d y}{d x}+y=x \log x

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

                \frac{d y}{d x}+Py= Q

                \begin{aligned} &P=\frac{1}{x^{\prime}} Q=\log x \\ &I f=e^{\int \frac{1}{x} d x} \\ &=e^{\log x} \\ &=x \end{aligned}

                \begin{aligned} &y I f=\int Q I f d x \\ &y x=\int x \log x d x+C \\ &=\log x \frac{x^{2}}{2}-\int \frac{1}{x} \frac{x^{2}}{2} d x+C \end{aligned}

                \begin{aligned} &x y=\frac{x^{2} \log x}{2}-\int \frac{x}{2} d x+C \\ &=\frac{x^{2} \log x}{2}-\frac{x^{2}}{4}+C \\ \end{aligned}

                 \begin{aligned} &=y=\frac{x \log x}{2}-\frac{x}{4}+\frac{C}{x} \\ &=4 x y=2 x^{2} \log x-x+C \end{aligned}

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