The differential equation of all parabolas whose axis is $y-\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{s}$ is

• Correct answer
  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

  $\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}=0$

Solution

Axis = $y$ axis
Vertex is (0, k)
Equation of parabola is
${\left(x-0\right)}^2=4a\left(y-k\right)$
$x^2=4ay-4ak$

Differentiate w.r.t x,
$2x=4a\frac{dy}{dx}$
$\Rightarrow$$x=2a\frac{dy}{dx}$
$\therefore \frac{1}{2a}=\frac{1}{x}\frac{dy}{dx}$
Differentiate w.r.t x,
$\frac{d}{dx}\left(\frac{1}{x}\cdot \frac{dy}{dx}\right)=\frac{d}{dx}\left(\frac{1}{2a}\right)$
$\Rightarrow$$\frac{1}{x}\cdot \frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}\left(-\frac{1}{x^2}\right)=0$
$\Rightarrow$$\therefore x\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}=0$

Hence, the correct answer is option: 'A'.