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Provide solution for RD Sharma maths class 12 chapter Differential Equations exercise 21.7 question 34

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Answer: \log |y|=\frac{2 x^{3}}{3}+x^{2}+2 x+2 \log (x-1)+c

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

Given: (x-1) \frac{d y}{d x}=2 x^{3} y

Solution: (x-1) \frac{d y}{d x}=2 x^{3} y

        \begin{aligned} &\Rightarrow \frac{d y}{y}=\frac{2 x^{3}}{(x-1)} d x \\\\ &\Rightarrow \frac{d y}{y}=\frac{2\left((x-1)\left(x^{2}+x+1\right)+1\right)}{(x-1)} d x \\\\ &\Rightarrow \frac{d y}{y}=2\left(x^{2}+x+1+\frac{1}{x-1}\right) d x \end{aligned}

          Integrating both sides

        \int \frac{d y}{y}=2\left[\int x^{2} d x+\int x d x+\int 1 d x+\int \frac{1}{x-1}\right] d x

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

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