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

Differentiating both sides with respect to x, we get:

$2 y \frac{d y}{d x}=a(-2 x)$

$\Rightarrow 2 y y^{\prime}=-2 a x$

$\Rightarrow y y^{\prime}=-\alpha x$            ...(1)

Again, differentiating both sides with respect to x, we get:

$y^{\prime} \cdot y^{\prime}+y y^{\prime \prime}=-a$

$\Rightarrow\left(y^{\prime}\right)^{2}+y y^{\prime \prime}=-a$                       ...(2)

Dividing equation (2) by equation (1), we get:

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

$\Rightarrow x y y^{\prime \prime}+x\left(y^{\prime}\right)^{2}-y y^{\prime \prime}=0$

This is the required differential equation of the given curve.