Question

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

Answer 1

Equation of such an ellipse is given by $\frac{x^2}{b^2}+\frac{y^2}{a^2}=0$

Here we have $a$ and $b$ as variables, so let's remove them to get the required differential equation.

Differentiating this on both sides, we get

$\frac{2x}{b^2}+\frac{{2yy}^{{}^{\prime }}}{a^2}=0$ --------(1)

Differentiating it again we will get

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

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

Putting this in equation (1), we get

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

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