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Please solve RD Sharma class 12 chapter Differential Equations exercise 21.7 question 29 maths textbook solution

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Answer: \log (x y)+x-\frac{y^{2}}{2}=C

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

Given: (y+x y) d x+\left(x-x y^{2}\right) d y=0


        \begin{aligned} &(y+x y) d x+\left(x-x y^{2}\right) d y=0 \\\\ &y(1+x) d x+x\left(1-y^{2}\right) d y=0 \\\\ &\frac{1+x}{x} d x+\frac{\left(1-y^{2}\right)}{y} d y=0 \end{aligned}

          Integrating both sides and also separate the terms

        \begin{aligned} &\int \frac{1}{x} d x+\int 1 d x+\int \frac{1}{y} d y-\int y d y=0 \\\\ &\log |x|+x+\log |y|-\frac{y^{2}}{2}=C \\\\ &\log (x y)+x-\frac{y^{2}}{2}=C \end{aligned}


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