From the following equation, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

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

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Solution

Given family is $y^{2} = a (b^{2} - x^{2}) . . . . . (i)$
On differentiating w.r.t. x, we get
$2 y y^{'} = a (0 - 2 x) \Rightarrow 2 y y^{'} = a (- 2 x) \Rightarrow y y^{'} = - a x . . . . . (i i)$
Again differentiating w.r.t. x, we get
$y y^{''} + (y^{'})^{2} = - a$ (using product rule of differentiation) .....(iii)
Now, for eleminating a, put the value of a Eq. (iii) in Eq. (ii), we get
$y y^{'} = [y y^{''} + (y^{'})^{2}] x \Rightarrow y y^{'} = x [(y^{'})^{2} + y y^{'}]$
which is the required differential equation.

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From the following equation, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.​ y2=a(b2−x2)

From the following equation, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

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