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

Toolbox:
• Product Rule: Differentiating a function which is in the form of product is $uv = uv' +vu'$
Step 1:
Given : $y^2 = a(b^2 - x^2)$
Differenting with respect to $x$ on both sides we get
$2yy' = -2ax$
Dividing on both sides by $-2$ we get,
$yy' = -ax$ ---------(1)
Step 2:
Differenting once again on both sides:
Using the information in the tool box we use product rule to differentiate yy'.
Let $y = u$ and $v = y'$. Hence $u' = y'$ and $v' = y''$
Applying this to differentiate equ (1) we get
$yy'' + y'.y' = -a$
$yy'' + (y')^2 = -a$ -------(2)
Step 3:
Dividing equ (2) by (1) we get
$\large\frac{[yy'' + (y')^2 = -a]}{[yy' = -ax]}$
We get $\large\frac{[yy'' + (y')^2]}{ yy' }=\frac{ 1}{x}$
on cross multiplying and rearranging we get
$xyy'' + x(y')^2 -yy'' = 0.$