The sum of  the order and the degree of differential equation of all the conics whose  standard axes coincide with the co-ordinate axes.
 

Any conic whose axes coincides with co-ordinate axis is,
$\begin{array}{l}{\mathrm{ax}}^2+{\mathrm{by}}^2=1\\ \Rightarrow 2\mathrm{ax}+2\mathrm{by}\cdot \mathrm{dy}/\mathrm{dx}=0\\ \Rightarrow \mathrm{ax}+\mathrm{bydy}/\mathrm{dx}=0\end{array}$
$\Rightarrow \mathrm{a}+\mathrm{b}\left(\mathrm{y}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}+{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right)=0$
From (i) and (ii),
$\begin{array}{l}\mathrm{a}=\frac{-\mathrm{by}}{\mathrm{x}}\cdot \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=-\mathrm{b}\left(\mathrm{y}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}+{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right)\\ \Rightarrow \frac{\mathrm{y}}{\mathrm{x}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=\mathrm{y}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}+{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\Rightarrow \mathrm{xy}\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)+\mathrm{x}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2-\mathrm{y}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=0\end{array}$