Question

The differential equation of all parabolas whose axis is $y$ -axis is

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

Answer 1

Answer: (A)

axis $=y$ axis
vertex is $(0, k )$
Equation of parabola is
$(x-0)^{2}=4 a(y-k)$
$x^{2}=4 a y-4 a k$
Differentiate w.r.t $x$
$2 x =4 a \frac{ dy }{ dx } $
$x =2 a \frac{ dy }{ dx }$
$\therefore \frac{1}{2 a }=\frac{1}{ x } \frac{ dy }{ dx }$
Differentiate w.r.t $x$,
$\frac{d}{d x}\left(\frac{1}{x} \cdot \frac{d y}{d x}\right)=\frac{d}{d x}\left(\frac{1}{2 a}\right) $
$\frac{1}{x} \cdot \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}\left(-\frac{1}{x^{2}}\right)=0 $
$\therefore x \frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}=0$