Question

The differential equation satisfied by the system of parabolas $y^2=4 a(x+a)$ is

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

Answer 1

Answer: (C)

$y^2=4 a x+4 a^2$
differentiate with respect to $x$
$\Rightarrow 2 y \frac{dy}{dx}=4 a $
$\Rightarrow a=\left(\frac{y}{2} \frac{dy}{dx}\right)$
so, required differential equation is
$y^2=\left(4 \times \frac{y}{2} \frac{d y}{d x}\right) x+4\left(\frac{y}{2} \frac{d y}{d x}\right)^2 $
$\Rightarrow y^2\left(\frac{d y}{d x}\right)^2+2 x y\left(\frac{d y}{d x}\right)-y^2=0$
$\Rightarrow y\left(\frac{d y}{d x}\right)^2+2 x\left(\frac{d y}{d x}\right)-y=0$