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Explain solution RD Sharma class 12 chapter Differential Equations exercise 21.7 question 32 maths

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Answer: \log (y+1)+x+\frac{x^{2}}{2}=c

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

Given: d y+(x+1)(y+1) d x=0

Solution: d y+(x+1)(y+1) d x=0

        \begin{aligned} &d y=-(x+1)(y+1) d x \\\\ &\frac{d y}{(y+1)}=-(x+1) d x \end{aligned}

          Integrating both sides

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

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