Find the differential equations of the family of all the ellipses having foci on the $y$-axis and centre at the origin.

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> [!TLDR]
>
> Differentiate the ellipse equation twice and eliminate both $a^2$ and $b^2$ exactly as Example 10.6 did for the foci-on-$x$-axis case; the algebra is symmetric in the two constants, so the same differential equation results.

**Step 1. Write the family.** $\dfrac{x^2}{b^2}+\dfrac{y^2}{a^2}=1$ ($a>b>0$, so the foci lie on the $y$-axis), with $a,b$ arbitrary.

**Step 2. Differentiate once.** $\dfrac{2x}{b^2}+\dfrac{2y}{a^2}\dfrac{dy}{dx}=0\ \Longrightarrow\ \dfrac{x}{b^2}+\dfrac{y}{a^2}\dfrac{dy}{dx}=0$. …(2)

**Step 3. Differentiate (2) again.** $\dfrac{1}{b^2}+\dfrac{1}{a^2}\left[y\dfrac{d^2y}{dx^2}+\left(\dfrac{dy}{dx}ight)^2ight]=0\ \Longrightarrow\ \dfrac{1}{b^2}=-\dfrac{1}{a^2}\left[y\dfrac{d^2y}{dx^2}+\left(\dfrac{dy}{dx}ight)^2ight]$. …(3)

**Step 4. Also solve (2) for $\dfrac1{b^2}$.** $\dfrac{1}{b^2}=-\dfrac{y}{x\,a^2}\dfrac{dy}{dx}$ (for $x
e0$). …

Q.Find the differential equations of the family of all the ellipses having foci on the $y$ -axis and centre at the origin.

Concept understanding — Formation of ODEs

Q.Find the differential equations of the family of all the ellipses having foci on the yyy-axis and centre at the origin.

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Q.Find the differential equations of the family of all the ellipses having foci on the $y$ -axis and centre at the origin.

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