Answer:
Clearly, the equation of given family of hyperbolas is \[\frac{{{y}^{2}}}{{{a}^{2}}}-\frac{{{x}^{2}}}{{{b}^{2}}}=1\] ?(i) where, a and o are arbitrary constants. Now, as there are two arbitrary constants, therefore we will differentiate Eq. (i) twice w.r.t. x. On differentiating Eq. (i) w.r.t. x, we ge \[\frac{2y}{{{a}^{2}}}\cdot \frac{dy}{dx}-\frac{1}{{{b}^{2}}}2x=0\] \[\Rightarrow \,\,\,\,\frac{y{{y}_{1}}}{{{a}^{2}}}-\frac{x}{{{b}^{2}}}=0\] ?(ii) Again differentiating Eq. (ii) w.r.t. x, we get \[\frac{1}{{{a}^{2}}}[y{{y}_{2}}+{{({{y}_{1}})}^{2}}]-\frac{1}{{{b}^{2}}}=0\] \[\Rightarrow \] \[\frac{1}{{{b}^{2}}}=\frac{1}{{{a}^{2}}}[y{{y}_{2}}+{{({{y}_{1}})}^{2}}]\] Putting the value of \[\frac{1}{{{b}^{2}}}\] in Eq. (ii), we get \[\frac{y{{y}_{1}}}{{{a}^{2}}}-\frac{1}{{{a}^{2}}}[y{{y}_{2}}+{{({{y}_{1}})}^{2}}]x=0\] \[\Rightarrow \] \[y{{y}_{1}}-xy{{y}_{2}}-x{{({{y}_{1}})}^{2}}=0\] \[\Rightarrow \] \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}+x{{\left( \frac{dy}{dx} \right)}^{2}}-y\left( \frac{dy}{dx} \right)=0,\] which is the required differential equation.
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