#### Please Solve R.D.Sharma class 12 Chapter 21  Differential Equations Exercise Revision Exercise Question 9 Maths textbook Solution.

$xy\frac{d^{2}y}{dx^{2}}+x\left ( \frac{dy}{dx} \right )^{2}-y\frac{dy}{dx}=0$

Hint:

You must know about the equation of ellipses

Given:

Family of ellipses having foci on y-axis and centre at origin

Solution:

Ellipse whose foci on y-axis

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$                              [Two constants, differentiate twice]

\begin{aligned} &\frac{d}{d x}\left[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\right]=\frac{d}{d x}(1) \\ &\frac{1}{a^{2}}\left(\frac{d\left(x^{2}\right)}{d x}\right)+\frac{1}{b^{2}}\left(\frac{d\left(y^{2}\right)}{d x}\right)=0 \\ &\frac{1}{a^{2}}(2 x)+\frac{1}{b^{2}}\left(2 y \frac{d y}{d x}\right)=0 \\ &\frac{2 x}{a^{2}}+\frac{2 y}{b^{2}}\left(\frac{d y}{d x}\right)=0 \\ &\frac{2 y}{b^{2}}\left(\frac{d y}{d x}\right)=-\frac{2 x}{a^{2}} \\ &\frac{y}{b^{2}}\left(\frac{d y}{d x}\right)=-\frac{x}{a^{2}} \\ &\frac{y}{x}\left(\frac{d y}{d x}\right)=-\frac{b^{2}}{a^{2}} \\ &\frac{y}{x} y^{\prime}=-\frac{b^{2}}{a^{2}} \end{aligned}

Again differentiate,