The Differential equation by eliminating the arbitrary constants from $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$ is

 

$$\begin{gathered} \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \\ diff. w.r.t. x \\ \Rightarrow \frac{2x}{a^2}-\frac{2y{y}^{\mathrm{\prime }}}{b^2}=0\to \left(1\right) \\ again diff. w.r.t. x \\ \Rightarrow \frac{2}{a^2}-\frac{2}{b^2}\left(y{y}^{\prime \prime }+y^{\mathrm{\prime }}{y}^{\mathrm{\prime }}\right)=0\to \left(2\right) \\ \text{ from (1)\& (2) } \\ xy{y}^{\prime \prime }+x{\left(y^{\mathrm{\prime }}\right)}^2-y{y}^{\mathrm{\prime }}=0 \end{gathered}$$