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Please solve RD Sharma class 12 chapter Differential Equations exercise 21.7 question 5 maths textbook solution

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Answer: \frac{y^{3}}{3}+\frac{y^{2}}{2}=\frac{x^{2}}{2}+\log |x|+c

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

Given: x y(y+1) d y=\left(x^{2}+1\right) d x

Solution:x y(y+1) d y=\left(x^{2}+1\right) d x

        \begin{aligned} &y(y+1) d y=\frac{\left(x^{2}+1\right) d x}{x} \\\\ &\left(y^{2}+y\right) d y=\left(\frac{x^{2}}{x}+\frac{1}{x}\right) d x \end{aligned}

        Integrating both sides

        \begin{aligned} &\int\left(y^{2}+y\right) d y=\int\left(x+\frac{1}{x}\right) d x \\\\ &\frac{y^{3}}{3}+\frac{y^{2}}{2}=\frac{x^{2}}{2}+\log |x|+c \end{aligned}

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