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  • Need solution for RD Sharma maths class 12 chapter 21 Differential Equation exercise Fill in the blank question 35

Need solution for RD Sharma maths class 12 chapter 21 Differential Equation exercise Fill in the blank question 35

Answers (1)

Answer:

 (x^{2}-y^{2})dy=2xydx

Hint:

 To remove the arbitrary constant we have differentiate the given equation.

Given:

 The differential equation of the family of curves x2 + y2 - 2ay = 0, where a is arbitrary constant is _____

Solution:

\begin{aligned} &x^{2}+y^{2}=2ay \\ &\Rightarrow \frac{x^{2}+y^{2}}{y}=2a \\ &\Rightarrow \frac{y\frac{\mathrm{d} y}{\mathrm{d} x}(x^{2}+y^{2})-(x^{2}+y^{2})\frac{\mathrm{d} y}{\mathrm{d} x}}{y^{2}}=0 \\ &\Rightarrow y\left [ 2x+2y\frac{\mathrm{d} y}{\mathrm{d} x} \right ]-(x^{2}+y^{2})\frac{\mathrm{d} y}{\mathrm{d} x}=0 \\ &\Rightarrow 2xy+2y^{2}\frac{\mathrm{d} y}{\mathrm{d} x}-(x^{2}+y^{2})\frac{\mathrm{d} y}{\mathrm{d} x}=0 \end{aligned}

\begin{aligned} &\Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x}(2y^{2}-x^{2}-y^{2})+2xy=0 \\ &\Rightarrow \frac{\mathrm{d} y}{\mathrm{d} x}(y^{2}-x^{2})+2xy=0 \\ &\Rightarrow (x^{2}-y^{2})\frac{\mathrm{d} y}{\mathrm{d} x}=2xy \\ &\Rightarrow (x^{2}-y^{2})dy=2xydx \end{aligned}

So, the answer is

\begin{aligned} & (x^{2}-y^{2})dy=2xydx \end{aligned}

Posted by

Gurleen Kaur

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