Question

The differential equation of all parabolas whose axes are parallel to y-axis, is

• A. $\raisebox{1ex}{${\mathrm{d}}^3\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{dx}}^3=0$}\right.$
• B. $\raisebox{1ex}{${\mathrm{d}}^2\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{dy}}^2=\mathrm{c}$}\right.$
• C. $\left(\raisebox{1ex}{${\mathrm{d}}^3\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{dx}}^3$}\right.\right)+\left(\raisebox{1ex}{${\mathrm{d}}^2\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{dy}}^2$}\right.\right)=0$
• D. $\left(\raisebox{1ex}{${\mathrm{d}}^2\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{dy}}^2$}\right.\right)+2\left(\raisebox{1ex}{$\mathrm{dy}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{dx}$}\right.\right)=\mathrm{c}$

Answer 1

The correct option is A

$\raisebox{1ex}{${\mathrm{d}}^3\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{dx}}^3=0$}\right.$

Explanation for the correct option:

Given a parabola whose axes are parallel to y-axis.

It means the vertex is not $(0,0).$

The equation becomes,

${\left(\mathrm{x}-\mathrm{h}\right)}^2=4\mathrm{b}\left(\mathrm{y}-\mathrm{k}\right)$

On differentiating,

$2\left(\mathrm{x}-\mathrm{h}\right)=4\mathrm{b}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)$

Again Differentiating, we get

$2=4\mathrm{b}\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)$

Again Differentiating, we get

$$\begin{gathered} 0=4\mathrm{b}\left(\frac{{\mathrm{d}}^3\mathrm{y}}{{\mathrm{dx}}^3}\right) \\ \left(\frac{{\mathrm{d}}^3\mathrm{y}}{{\mathrm{dx}}^3}\right)=0 \end{gathered}$$

Hence, the correct answer is option (A).