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Need solution for RD Sharma maths class 12 chapter Differential Equations exercise 21.7 question 51

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Answer: 2 x^{2} y+y=1

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

Given: \frac{d y}{d x}=-4 x y^{2}, y=1 \text { when } x=0


        \begin{aligned} &\frac{d y}{d x}=-4 x y^{2} \\ &\Rightarrow \frac{d y}{y^{2}}=-4 x d x \end{aligned}

          Integrating both sides

        \begin{aligned} &\int \frac{1}{y^{2}}=-4 \int x d x \\\\ &\Rightarrow \frac{y^{-2+1}}{-2+1}=\frac{-4 x^{2}}{2}+c \\\\ &\Rightarrow \frac{-1}{y}=-2 x^{2}+c \\ &\Rightarrow-\frac{1}{y}=-2 x^{2}+c \end{aligned}                ...............(1)

         Now y=1 when x=0


        Put in (1)

        \begin{aligned} \Rightarrow \frac{-1}{y} &=-2 x^{2}-1 \end{aligned}

        \Rightarrow 2 x^{2} y+y=1



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