Advertisements
Advertisements
Question
Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin
Advertisements
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.
APPEARS IN
RELATED QUESTIONS
In the following example verify that the given expression is a solution of the corresponding differential equation:
y = e-x + Ax + B; `"e"^"x" ("d"^2"y")/"dx"^2 = 1`
Solve the following differential equation:
`(cos^2y)/x dy + (cos^2x)/y dx` = 0
For the following differential equation find the particular solution satisfying the given condition:
`cos("dy"/"dx") = "a", "a" ∈ "R", "y"(0) = 2`
Choose the correct option from the given alternatives:
The solution of `("x + y")^2 "dy"/"dx" = 1` is
Choose the correct option from the given alternatives:
The solution of the differential equation `"dy"/"dx" = sec "x" - "y" tan "x"`
Choose the correct option from the given alternatives:
`"x"^2/"a"^2 - "y"^2/"b"^2 = 1` is a solution of
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
`"y"^2 = "a"("b - x")("b + x")`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = a sin (x + b)
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = `sqrt("a" cos (log "x") + "b" sin (log "x"))`
Solve the following differential equation:
x dy = (x + y + 1) dx
Find the particular solution of the following differential equation:
`("x + 2y"^2) "dy"/"dx" = "y",` when x = 2, y = 1
Find the particular solution of the following differential equation:
`"dy"/"dx" - 3"y" cot "x" = sin "2x"`, when `"y"(pi/2) = 2`
Select and write the correct alternative from the given option for the question
The solution of `("d"y)/("d"x)` = 1 is
Select and write the correct alternative from the given option for the question
The solutiion of `("d"y)/("d"x) + x^2/y^2` = 0 is
Form the differential equation of family of standard circle
Find the differential equation of the family of all the parabolas with latus rectum 4a and whose axes are parallel to the x-axis
If m and n are respectively the order and degree of the differential equation of the family of parabolas with focus at the origin and X-axis as its axis, then mn - m + n = ______.
The differential equation for all the straight lines which are at the distance of 2 units from the origin is ______.
For the curve C: (x2 + y2 – 3) + (x2 – y2 – 1)5 = 0, the value of 3y' – y3 y", at the point (α, α), α < 0, on C, is equal to ______.
A particle is moving along the X-axis. Its acceleration at time t is proportional to its velocity at that time. Find the differential equation of the motion of the particle.
