Question

Form the differential equation corresponding to y² = a (b − x²) by eliminating a and b.

Answer 1

The equation of the family of curves is
$y^2=a\left(b-x^2\right)$, ...(1)
where $a\mathrm{and}b$ are parameters.
This equation contains two arbitrary constants, so we shall get a differential equation of second order.
Differentiating equation (1) with respect to $x$, we get
$2y\frac{dy}{dx}=-2ax$ ...(2)
Differentiating equation (2) with respect to $x$, we get
${\left(\frac{dy}{dx}\right)}^2+y\frac{d^{2}y}{d{x}^2}=-a$ ...(3)
From (2) and (3), we get
$y\frac{dy}{dx}=x\left[{\left(\frac{dy}{dx}\right)}^2+y\frac{d^{2}y}{d{x}^2}\right]$

It is the required differential equation.