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Provide solution for RD Sharma maths class 12 chapter Differential Equations exercise 21.7 question 2

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Answer: C \sqrt{1+x^{2}}

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

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

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

        \begin{aligned} &\frac{d y}{y}=\frac{x d x}{1+x^{2}} \\\\ &\text { Put } 1+x^{2}=t \\\\ &2 x d x=d t \end{aligned}

        \begin{aligned} &x d x=\frac{d t}{2} \\\\ &\frac{d y}{y}=\frac{d t}{2 t} \end{aligned}

Integrating both sides

        \begin{aligned} &\log |y|=\frac{1}{2} \log |t|+\log C \\\\ &\log |y|=\frac{1}{2} \log \left|1+x^{2}\right|+\log C \end{aligned}

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


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