Question

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

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

Answer 1

Answer: (C)

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