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

$\begin{array}{l}Equationofparabola\\ {(x-h)}^2=4a(y-k)\\ \Rightarrow 2(x-h)=4a\left(\frac{dy}{dx}\right)\\ Againdiffw.r.to\\ \Rightarrow 2=4a\frac{d^{2}y}{d{x}^2}\\ \Rightarrow 1=2a\frac{d^{2}y}{d{x}^2}\\ Againdiffw.r.to\text{'}x\text{'}\\ \Rightarrow 0=2a\frac{d^{3}y}{d{x}^3}\\ \Rightarrow \frac{d^{3}y}{d{x}^3}=0\end{array}$