#### Explain solution RD Sharma class 12 chapter Differential Equation exercise 21.2 question 18 maths

The required differential equation is

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

Hint:

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

Given:

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

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

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

Substituting this value of 1/a2 in equation (ii), we get

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

The required differential equation is

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