The form of the differential equation of the central conics, is

Central conics are given by

$\begin{array}{l}a{x}^2+b{y}^2=1\\ \Rightarrow 2ax+2by{y}_1=0\\ \Rightarrow ax+by{y}_1=0\\ \Rightarrow \frac{-a}{b}=\frac{y{y}_1}{x}\end{array}$

Differentiating $ax+by{y}_1=0$ with respect to x, we' obtain

$\begin{array}{r}a+b{y_1}^2+by{y}_2=0\\ \Rightarrow \frac{-a}{b}={y_1}^2+y{y}_2\end{array}$

From (i) and (ii), we get 

$\frac{y{y}_1}{x}=y_1^2+y{y}_{1/2}\Rightarrow y{y}_1=x{y}_1^2+xy{y}_2$