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

let $\frac{dy}{dx}=y^{\prime }\text{and}\frac{d^{2}y}{d{x}^2}=y"$

Let the curve

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

Differentiating both sides w.r.t. x
we get,

$2y.\frac{dy}{dx}=0-2ax$

$2y{y}^{\prime }=-2ax$

$y{y}^{\prime }=-ax$

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

Now,

$y{y}^{\prime }=-ax$

Again, differentiating both sides w.r.t. x
we get,
Product rule$(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$

$\frac{dy}{dx}.y^{\prime }+y.\frac{d\left(y^{\prime }\right)}{dx}=-a\frac{dx}{dx}$

$y^{\prime }\times y^{\prime }+y\times y"=-a$

$y^{\prime }2+yy"=-\left(\frac{-y}{x}{y}^{\prime }\right)$

$y^{\prime 2}+yy"=\frac{y{y}^{\prime }}{x}$

$x{y}^{\prime 2}+xyy"=y{y}^{\prime }$

$x{y}^{\prime 2}+xyy"=y{y}^{\prime }$

$x{y}^{\prime 2}+xyy"-y{y}^{\prime }=0$

Final Answer:
Hence, the required differential equation is

$x{y}^{\prime 2}+xyy"-y{y}^{\prime }=0$