#### Provide solution for RD Sharma maths class 12 chapter 21 Differential Equation exercise Fill in the blank question 10

2

Hint:

Differentiating w.r.t to ‘x’

Given:

The family of ellipses having foci on x-axis and center at the origin, is given by

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$

Solution:

Differentiating w.r.t ‘x’, we get

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

Again by differentiating w.r.t ‘x’ we get

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

The required equation is

\begin{aligned} &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} \qquad \qquad \dots(i) \end{aligned}

∴ Order of the differential equation is the highest derivative present in the differential equation(i)

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