Question

The differential equations of all conics whose axes coincide with the co-ordinate axis

• A. $x y \frac{d^{2}y}{dx^{2}}+x\left(\frac{dy}{dx}\right)^{2}+ y \frac{dy}{dx} = 0$
• B. $x y \frac{d^{2}y}{dx^{2}}+x\left(\frac{dy}{dx}\right)^{2}+x \frac{dy}{dx} = 0$
• C. $x y \frac{d^{2}y}{dx^{2}}+x\left(\frac{dy}{dx}\right)^{2}- y \frac{dy}{dx} = 0$
• D. $x y \frac{d^{2}y}{dx^{2}}-x\left(\frac{dy}{dx}\right)^{2}+ y \frac{dy}{dx} = 0$

Answer 1

Answer: (C)

Any conic whose axes coincide with co-ordinate axis is
$ax^{2} + by^{2} = 1\quad ..\left(i\right)$
Diff. both sides w.r.t. 'x', we get
$2ax+2by \frac{dy}{dx} = 0$ i.e. $ax +by \frac{dy }{dx} = 0 \quad ..\left(ii\right)$
Diff. again, $a + b \left(y \frac{d^{2}y}{dx^{2}}+\left(\frac{dy}{dx}\right)^{2}\right) = 0\quad ..\left(iii\right)$
From $\left(ii\right), \frac{a}{b} = -\frac{ydy / dx}{x}$
From $\left(iii\right), \frac{a}{b} = -\left(y \frac{d^{2}y}{dx^{2}}+\left(\frac{dy}{dx}\right)^{2}\right)$
$\therefore \quad \frac{y \frac{dy}{dx}}{x} = y \frac{d^{2}y}{dx^{2}}+\left(\frac{dy}{dx}\right)^{2}$
$\Rightarrow \quad x y \frac{d^{2}y}{dx^{2}}+x\left(\frac{dy}{dx}\right)^{2}- y \frac{dy}{dx} = 0$