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

$(x^{2}-y^{2})dy=2xydx$

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}

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