#### Need solution for RD Sharma maths class 12 chapter Differential Equation exercise 21.2 question 17

The required differential equation is

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

Hint:

Ellipse centre at the origin and foci on the x-axis

Given:

Ellipse having the centre at the origin and foci on the x-axis

Solution:

Equation of required ellipse is

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \qquad \qquad \dots (i)$

Where a and b are arbitrary constants

Differentiating equation (i) with respect to x, we get

\begin{aligned} &\frac{2x}{a^{2}}+\frac{2y}{b^{2}}\frac{\mathrm{d} y}{\mathrm{d} x}=0 \\ &\frac{2x}{a^{2}}=-\frac{2y}{b^{2}}\frac{\mathrm{d} y}{\mathrm{d} x} \\ &\frac{-b^{2}}{a^{2}}=\frac{y}{x}\frac{\mathrm{d} y}{\mathrm{d} x} \qquad \qquad \dots(ii) \end{aligned}

Differentiating equation (ii) with respect to x, we get

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

The required differential equation is

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