Question

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

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

Answer 1

Answer: (A)

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