Question

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b: $y^2 = a (b^2 - x^2)$

Answer 1

To form a differential equation representing the given family of curves by eliminating arbitrary constants a and b: $y^2 = a (b^2 - x^2)$,we must differentiate the expression with respect to x and replace the values of arbitrary constants

${y}^{2} = a( {b}^{2} - {x}^{2} ) \\ \\ {y}^{2} = a {b}^{2} - a {x}^{2} \\ \\ 2y \frac{dy}{dx} = 0 - 2ax \\ \\ y \frac{dy}{dx} = - ax \: \: \: \: \: eq1 \\ \\ differentiate \: again \\ \\ y \frac{ {d}^{2}y }{ {dx}^{2} } + \bigg({ \frac{dy}{dx} }\bigg)^{2} = - a$
put value of -a from eq1

$y \frac{ {d}^{2}y }{ {dx}^{2} } +\bigg({ \frac{dy}{dx} }\bigg)^{2} = \frac{y}{x} \frac{dy}{dx} \\ \\ y \frac{ {d}^{2}y }{ {dx}^{2} } + \bigg({ \frac{dy}{dx} }\bigg)^{2} - \frac{y}{x} \frac{dy}{dx} = 0$
is the required differential equation.