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Please solve RD Sharma class 12 chapter Differential Equations exercise 21.11 question 21 maths textbook solution

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Answer: 9 y+4 x^{2}=0

Given: Curve passes through \left ( 3,-4 \right ) and given slope of curve =\frac{2y}{x}

To find: We have to show that the equation which satisfies the curve.

Hint: Solve of curve =\frac{2 y}{x}, \frac{d y}{d x}=\frac{2 y}{x}, solve this to find the equation of the curve.

Solution: Slope of curve =\frac{2 y}{x}

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

Integrating on both sides

        \begin{aligned} &=\int \frac{d y}{y}=2 \int \frac{d x}{x} \\\\ &=\log y=2 \log x+\log C \\\\ &=y=x^{2} C \ldots(i) \end{aligned}

As it passes through \left ( 3,-4 \right )

        \begin{aligned} &=-4=(3)^{2} C \\\\ &=-4=9 C \\\\ &=C=-\frac{4}{9} \end{aligned}

So equation (i) becomes

        \begin{aligned} &=y=-\frac{4}{9} x^{2} \\\\ &=9 y=-4 x^{2} \\\\ &=9 y+4 x^{2}=0 \end{aligned}



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