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,

\[\alpha\text{ and }\beta\] are two arbitrary constants.

Differentiating (1) with respect to x, we get

\[2\left( x - \beta \right) = 4a\frac{dy}{dx}\]

\[ \Rightarrow 1 = 2a\frac{d^2 y}{d x^2}\]

\[ \Rightarrow 0 = 2a\frac{d^3 y}{d x^3}\]

\[ \Rightarrow \frac{d^3 y}{d x^3} = 0\]