Form the differential equation of the family of ellipses having foci on y-axis and centre at origin.

The equation of the family of ellipses having foci on the y-axis and the centre at origin is as follows:

$\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1$              ...(1)

Differentiating equation (1) with respect to x, we get:

$\frac{2 x}{b^{2}}+\frac{2 y y^{\prime}}{b^{2}}=0$

$\Rightarrow \frac{x}{b^{2}}+\frac{y y^{\prime}}{a^{2}}=0$           ...(2)

Again, differentiating with respect to x, we get:

$\frac{1}{b^{2}}+\frac{y^{\prime} \cdot y^{\prime}+y \cdot y^{\prime \prime}}{a^{2}}=0$

$\Rightarrow \frac{1}{b^{2}}+\frac{1}{a^{2}}\left(y^{\prime 2}+y y^{\prime \prime}\right)=0$

$\Rightarrow \frac{1}{b^{2}}=-\frac{1}{a^{2}}\left(y^{\prime 2}+y y^{\prime \prime}\right)$

Substituting this value in equation (2), we get:

$x\left[-\frac{1}{a^{2}}\left(\left(y^{\prime}\right)^{2}+y y^{\prime \prime}\right)\right]+\frac{y y^{\prime}}{a^{2}}=0$

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

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

This is the required differential equation.