Find the differential equation of all ellipses whose centre is origin and foci is on the x-axis.

Explanation

The equation of family of such ellipses is `(x^(2))/(a^(2)) + (y^(2))/(b^(2)) = 1 (a gt b)`
where, a, b are arbitrary constants. `(x^(2))/(a^(2)) + (y^(2))/(b^(2)) = 1`. ...(1)
Diferentiating with respect to x we get
`(2x)/(a^(2)) + (2y)/(b^(2)).y' - 0`
`rArr (x)/(a^(2)) + (y y')/(b^(2)) = 0` ...(2)
Differentiating again w.r.t. x, we get
`(1)/(a^(2)) + (1)/(b^(2))(y y'' + y^('2)) = 0` ...(3)
From (2) `(1)/(a^(2)) = -(y y')/(x b^(2))`
Substituting in (3)
`(-y y')/(x b^(2)) + (1)/(b^(2)) (y y'' + y^('2)) = 0`
`rArr -y y' + x y y'' + x y^(2) = 0`
`rArr x y y' - y y' + xy^('2) = 0`
This is the required differential equation.