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Q.
The differential equation representing the family of hyperbolas $a^{2} x^{2}-b^{2} y^{2}=c^{2}$ is
Differential Equations
Solution:
Differentiating the equation twice w.r.t. $x,$ we get
$2 a^{2} x-2 b^{2} y y'=0, a^{2}-b^{2}\left(y'^{2}+y y''\right)=0$
Eliminating $a^{2}$ and $b^{2},$ we get the differential equation
$\frac{y''}{y'}+\frac{y'}{y}=\frac{1}{x}$