Question

Form a differential equation representing the given family
of curves by eliminating arbitrary constant a and b.

$y^2=a\left(b^2-x^2\right)$

Answer 1

$y^2=a(b^2-x^2)$

Differentiating we get

$2y{y}^{{}^{\prime }}=-2ax$..............(i)

$a=\frac{-y{y}^{{}^{\prime }}}{x}$

Differentiating (i) we get,

$2y^{{}^{\prime }2}+2y{y}^{{}^{\prime \prime }}=-2a$

$2y^{{}^{\prime }2}+2y{y}^{{}^{\prime \prime }}=\frac{y{y}^{{}^{\prime }}}{x}$ is the required differential equation.