Advertisements
Advertisements
प्रश्न
Find the differential equation all parabolas having a length of latus rectum 4a and axis is parallel to the axis.
Advertisements
उत्तर
Let A(h, k) be the vertex of the parabola whose length of the latus rectum is 4a.
Then the equation of the parabola is
(y - k)2 = 4a(x - h), where h and k are arbitrary constants. Differentiating w.r.t. x, we get
`2("y - k") * "d"/"dx" ("y - k") = 4"a" "d"/"dx" ("x - h")`
∴ `2("y - k")("dy"/"dx" - 0) = "4a"(1 - 0)`
∴ `2("y - k")"dy"/"dx" = "4a"`
∴ `("y - k")"dy"/"dx" = "2a"` ...(1)
Differentiating again w.r.t. x, we get
`("y - k") * "d"/"dx" ("dy"/"dx") + "dy"/"dx" * "d"/"dx" ("y - k") = 0`
∴ `("y - k")("d"^2"y")/"dx"^2 + "dy"/"dx" * ("dy"/"dx" - 0) = 0`
∴ `("y - k")("d"^2"y")/"dx"^2 + ("dy"/"dx")^2 = 0`
∴ `"2a"/(("dy"/"dx")) * ("d"^2"y")/"dx"^2 + ("dy"/"dx")^2 = 0` ....[By (1)]
∴ `"2a" ("d"^2"y")/"dx"^2 + ("dy"/"dx")^3 = 0`
This is the required D.E.
APPEARS IN
संबंधित प्रश्न
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
x3 + y3 = 4ax
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = A cos (log x) + B sin (log x)
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y2 = (x + c)3
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = Ae5x + Be-5x
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = 4(x - b)
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = a + `"a"/"x"`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
c1x3 + c2y2 = 5
Find the differential equation of the ellipse whose major axis is twice its minor axis.
Find the differential equation of all circles having radius 9 and centre at point (h, k).
Form the differential equation of family of lines parallel to the line 2x + 3y + 4 = 0.
Form the differential equation of all parabolas whose axis is the X-axis.
In the following example verify that the given expression is a solution of the corresponding differential equation:
y = xm; `"x"^2 ("d"^2"y")/"dx"^2 - "mx" "dy"/"dx" + "my" = 0`
In the following example verify that the given expression is a solution of the corresponding differential equation:
y = `"a" + "b"/"x"; "x" ("d"^2"y")/"dx"^2 + 2 "dy"/"dx" = 0`
In the following example verify that the given expression is a solution of the corresponding differential equation:
y = `"e"^"ax"; "x" "dy"/"dx" = "y" log "y"`
Solve the following differential equation:
`"dy"/"dx" = (1 + "y")^2/(1 + "x")^2`
Solve the following differential equation:
`"sec"^2 "x" * "tan y" "dx" + "sec"^2 "y" * "tan x" "dy" = 0`
Solve the following differential equation:
`"dy"/"dx" = - "k",` where k is a constant.
Solve the following differential equation:
`"dy"/"dx" = "e"^("x + y") + "x"^2 "e"^"y"`
For the following differential equation find the particular solution satisfying the given condition:
3ex tan y dx + (1 + ex) sec2 y dy = 0, when x = 0, y = π.
For the following differential equation find the particular solution satisfying the given condition:
`(e^y + 1) cos x + e^y sin x. dy/dx = 0, "when" x = pi/6,` y = 0
Reduce the following differential equation to the variable separable form and hence solve:
`"x + y""dy"/"dx" = sec("x"^2 + "y"^2)`
Reduce the following differential equation to the variable separable form and hence solve:
(2x - 2y + 3)dx - (x - y + 1)dy = 0, when x = 0, y = 1.
Choose the correct option from the given alternatives:
x2 + y2 = a2 is a solution of
Choose the correct option from the given alternatives:
The differential equation of all circles having their centres on the line y = 5 and touching the X-axis is
Choose the correct option from the given alternatives:
`"x"^2/"a"^2 - "y"^2/"b"^2 = 1` is a solution of
In the following example verify that the given function is a solution of the differential equation.
`"x"^2 = "2y"^2 log "y", "x"^2 + "y"^2 = "xy" "dx"/"dy"`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = `sqrt("a" cos (log "x") + "b" sin (log "x"))`
Form the differential equation of all the lines which are normal to the line 3x + 2y + 7 = 0.
Solve the following differential equation:
`"dy"/"dx" = "x"^2"y" + "y"`
The general solution of `(dy)/(dx)` = e−x is ______.
Find the differential equation from the relation x2 + 4y2 = 4b2
The differential equation having y = (cos-1 x)2 + P (sin-1 x) + Q as its general solution, where P and Q are arbitrary constants, is
The family of curves y = `e^("a" sin x)`, where a is an arbitrary constant, is represented by the differential equation.
Find the differential equation of the family of circles passing through the origin and having their centres on the x-axis
Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin
The differential equation of all lines perpendicular to the line 5x + 2y + 7 = 0 is ____________.
The general solution of the differential equation of all circles having centre at A(- 1, 2) is ______.
The elimination of the arbitrary constant m from the equation y = emx gives the differential equation ______.
The differential equation representing the family of parabolas having vertex at origin and axis along positive direction of X-axis is ______.
The differential equation for all the straight lines which are at the distance of 2 units from the origin is ______.
Form the differential equation of all lines which makes intercept 3 on x-axis.
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 ______.
If y = (tan–1 x)2 then `(x^2 + 1)^2 (d^2y)/(dx^2) + 2x(x^2 + 1) (dy)/(dx)` = ______.
Form the differential equation whose general solution is y = a cos 2x + b sin 2x.
Find the particular solution of the differential equation `x^2 dy/dx + y^2 = xy dy/dx`, if y = 1 when x = 1.
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.
