Question

Show that the differential equation of all parabolas which have their axes parallel to y-axis is $\frac{d^{3}y}{d{x}^3}=0$.

Answer 1

The equation of the family of parabolas axis parallel to y-axis is given by
${\left(x-\beta \right)}^2=4a\left(y-\alpha \right)$ .....(1)
Here, $\alpha \mathrm{and}\beta$ are two arbitrary constants.

Differentiating (1) with respect to x, we get
$$\begin{gathered} 2\left(x-\beta \right)=4a\frac{dy}{dx} \\ \Rightarrow 1=2a\frac{d^{2}y}{d{x}^2} \\ \Rightarrow 0=2a\frac{d^{3}y}{d{x}^3} \\ \Rightarrow \frac{d^{3}y}{d{x}^3}=0 \end{gathered}$$