The differential equation of all parabolas having their axis of symmetry with the axis of x is

$\begin{array}{l}Equationofparabolais\\ y^2=4a(x-h)\\ Numberofarbitrary\mathrm{costan}t=2\\ \Rightarrow 2y\frac{dy}{dx}=4a(1-0)\\ \Rightarrow y\frac{dy}{dx}=2a\\ Againdiffw.r.to\text{'}x\text{'}\\ \Rightarrow y.\frac{d^{2}y}{d{x}^2}+\left(\frac{dy}{dx}\right)\left(\frac{dy}{dx}\right)=0\\ \Rightarrow y.\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2=0\end{array}$