Question

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

Answer 1

The equation of the ellipses having foci on y-axis and centre at the origin is given by

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

Here,

b > a

Since these are two parameters, so we differentiate the equation twice.

Differentiating with respect to x, we get

\[\frac{2x}{a^2} + \frac{2y}{b^2}y' = 0\]
\[ \Rightarrow \frac{x}{a^2} + \frac{y}{b^2}y' = 0 . . . . . \left( 2 \right)\]
\[ \Rightarrow \frac{1}{a^2} + \frac{1}{b^2} \left( y' \right)^2 + \frac{y}{b^2}y'' = 0 . . . . . \left( 3 \right)\]
Multiplying throughout by x, we get
\[\frac{x}{a^2} + \frac{x}{b^2} \left( y' \right)^2 + \frac{xy}{b^2}y'' = 0 . . . . . \left( 4 \right)\]
Subtracting (2) from (4), we get
\[\frac{1}{b^2}\left[ x \left( y' \right)^2 + xyy'' - yy' \right] = 0 \]
\[ \Rightarrow x \left( y' \right)^2 + xyy'' - yy' = 0\]