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Provide solution for RD Sharma maths class 12 chapter Differential Equations exercise 21.7 question 45 sub question (ii)

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Answer: \log |y|=\frac{5}{2} \log |x|

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

Given: 2 x \frac{d y}{d x}=5 y, y(1)=1


        \begin{aligned} &2 x \frac{d y}{d x}=5 y \\\\ &\Rightarrow \frac{d y}{5 y}=\frac{d x}{2 x} \end{aligned}

        Integrating both sides

        \Rightarrow \int \frac{d y}{5 y}=\int \frac{d x}{2 x} \Rightarrow \frac{1}{5} \log |y|=\frac{1}{2} \log |x|+c        ...................(1)

        Now given that y(1)=1 \quad \therefore y=1 \text { at } x=1

        \therefore \frac{1}{5} \log |y|=\frac{1}{2} \log |x|+c \Rightarrow c=0[\therefore \log 1=0]

        Put in (1)

        \frac{1}{5} \log |y|=\frac{1}{2} \log |x|+0 \Rightarrow \log |y|=\frac{5}{2} \log |x|

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