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Question
Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin
Solution
The equation of the family of ellipses having centre at the origin and foci on the y-axis, is given by `x^2/"a"^2 + y^2/"b"^2` = 1 .......(1)
where b > a and a, b are the parameters or a,b are arbitrary constant.
Differentiating equation (1) twice successively, because we have two arbitrary constant, we get
`(2x)/"a"^2 + (2y)/"b"^2 ("d"y)/("d"x)` = 0
`2(x/"a"^2 + y/"b"^2 ("d"y)/("d"x))` = 0
`x/"a"^2 + y/"b"^2 ("d"y)/("d"x)` = 0 .......(2)
Again differentiating equation 2) w.r.t x,
`1/"a"^2 + y/"b"^2 ("d"^2y)/("d"x^2) + ("d"y)/("d"x) ("d"y)/("d"x "b"^2)` = 0
`1/"a"^2 + y/"b"^2 ("d"^2y)/("d"x^2) + (("d"y)/("d"x))^2 1/"b"^2` = 0
Multiply by x
`x/"a"^2 + x/"b"^2 (("d"^2y)/("d"x^2)) + (("d"y)/("d"x))^2 x/"b"^2` = 0 .......(3)
Equation (3) – (2) we get
`x/"a"^2 + (xy)/"b"^2 (("d"^2y)/("d"x^2)) + (("d"y)/("d"x))^2 (x/"b"^2) - (x/"a"^2 + y/"b"^2 ("d"y)/("d"x))` = 0
`(xy)/"b"^2 (("d"^2y)/("d"x^2)) + (("d"y)/("d"x))^2 x/"b"^2 - y/"b"^2 ("d"y)/("d"x)` = 0
Taking `1/"b"^2` outside, we get
`1/"b"^2 [xy ("d"^2y)/("d"x^2) + x(("d"y)/("d"x))^2 - y("d"y)/("d"x)]` = 0
`xy ("d"^2y)/("d"x^2) + x(("d"y)/("d"x))^2 - y("d"y)/("d"x)` = 0 is the required differential equation.
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