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Explain solution RD Sharma class 12 chapter Differential Equations exercise 21.7 question 40 maths

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Answer: 3 \sqrt{\frac{y}{2}}=\sqrt{x}

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

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

Solution:2 x \frac{d y}{d x}=3 y

        \frac{d x}{3 y}=\frac{d x}{2 x}

        Integrating both sides

        \begin{aligned} &\int \frac{d x}{3 y}=\int \frac{d x}{2 x} \\\\ &\frac{1}{3} \log |y|=\frac{1}{2} \log |x|+c \end{aligned}

        \begin{aligned} &\text { Put } y(1)=2 \text { where, } y=2 \& x=1 \\\\ &\frac{1}{3} \log |2|=\frac{1}{2} \log |1|+c \\\\ &\frac{1}{3} \log |2|=c \end{aligned}

        \begin{aligned} &{[\log 1=0]} \\\\ &\frac{1}{3} \log |y|=\frac{1}{2} \log |x|+\frac{1}{3} \log |2| \\\\ &\frac{1}{3} \log |y|-\frac{1}{3} \log |2|=\frac{1}{2} \log |x| \end{aligned}

        \begin{aligned} &\frac{1}{3} \log \frac{y}{2}=\frac{1}{2} \log |x| \\\\ &\log \sqrt[3]{\frac{y}{2}}=\log \sqrt{x} \\\\ &\sqrt[3]{\frac{y}{2}}=\sqrt{x} \end{aligned}

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