#### Please solve RD Sharma class 12 chapter Differential Equation exercise 21.2 question 6 maths textbook solution

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

Hint:

Differentiating the given equation, let it equation (i) and substitute a from (iii) in (ii)

Given:

$y^{2}=a(b-x^{2})$

Solution:

$y^{2}=a(b-x^{2})$

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

Again,

\begin{aligned} &\frac{\mathrm{d} }{\mathrm{d} x}\left ( y\frac{\mathrm{d} y}{\mathrm{d} x}\right ) =\frac{\mathrm{d} }{\mathrm{d} x}(-ax) \\ &\left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^{2}+y\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}=-a \qquad \dots(iii) \end{aligned}

Substitute equation (iii) in (ii),

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