Advertisements
Advertisements
प्रश्न
Find the differential equation of all circles having radius 9 and centre at point (h, k).
Advertisements
उत्तर
Equation of the circle having radius 9 and centre at point (h, k) is
(x - h)2 + (y - k)2 = 81, .....(1)
where h and k are arbitrary constant.
Differentiating (1) w.r.t. x, we get
`2("x - h") * "d"/"dx" ("x - h") + 2 ("y - k") * "d"/"dx" ("y - k") = 0`
∴ (x - h)(1 - 0) + (y - k)`("dy"/"dx" - 0) = 0`
∴ (x - h) + (y - k) `"dy"/"dx" = 0` .....(2)
Differentiating again w.r.t. x, we get
`"d"/"dx" ("x - h") + ("y - k") * "d"/"dx"("dy"/"dx") + "dy"/"dx" * "d"/"dx" ("y - k") = 0`
∴ `(1 - 0) + ("y - k") ("d"^2"y")/"dx"^2 + "dy"/"dx" * ("dy"/"dx" - 0) = 0`
∴ `("y - k") ("d"^2"y")/"dx"^2 + ("dy"/"dx")^2` + 1 = 0
∴ `("y - k") ("d"^2"y")/"dx"^2 = - [("dy"/"dx")^2 + 1]`
∴ `"y - k" = (- ("dy"/"dx")^2 + 1)/(("d"^2"y")/"dx"^2` ....(3)
From (2), x - h = - (y - k)`"dy"/"dx"`
Substituting the value of (x - h) in (1), we get
`("y - k")^2 ("dy"/"dx")^2 + ("y - k")^2 = 81`
∴ `("dy"/"dx")^2 + 1 = 81/("y - k")^2`
∴ `("dy"/"dx")^2 + 1 = (81 * ("d"^2"y")/"dx"^2)/[("dy"/"dx")^2 + 1]^2`
∴ `81 (("d"^2"y")/"dx"^2)^2 = [("dy"/"dx")^2 + 1]^3`
This is the required D.E.
APPEARS IN
संबंधित प्रश्न
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:
y = Ae5x + Be-5x
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:
y = c1e2x + c2e5x
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
c1x3 + c2y2 = 5
Form the differential equation of family of lines parallel to the line 2x + 3y + 4 = 0.
Solve the following differential equation:
`"y" - "x" "dy"/"dx" = 0`
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:
`y(1 + log x) dx/dy - x log x = 0, y = e^2,` when x = e
Reduce the following differential equation to the variable separable form and hence solve:
`"dy"/"dx" = cos("x + y")`
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:
`cos^2 ("x - 2y") = 1 - 2 "dy"/"dx"`
Solve the following differential equation:
(x2 + y2)dx - 2xy dy = 0
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
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 = `"Ae"^(3"x" + 1) + "Be"^(- 3"x" + 1)`
Form the differential equation of all parabolas which have 4b as latus rectum and whose axis is parallel to the Y-axis.
Form the differential equation of all the lines which are normal to the line 3x + 2y + 7 = 0.
Form the differential equation of the hyperbola whose length of transverse and conjugate axes are half of that of the given hyperbola `"x"^2/16 - "y"^2/36 = "k"`.
Solve the following differential equation:
`"dy"/"dx" + "y cot x" = "x"^2 "cot x" + "2x"`
The general solution of `(dy)/(dx)` = e−x 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
Find the general solution of `("d"y)/("d"x) = (1 + y^2)/(1 + x^2)`
Find the differential equation of family of all ellipse whose major axis is twice the minor axis
Find the differential equation by eliminating arbitrary constants from the relation y = (c1 + c2x)ex
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 all non-vertical lines in a plane
Find the differential equation of the family of all non-horizontal lines in a plane
Find the differential equation of the family of parabolas with vertex at (0, –1) and having axis along the y-axis
Choose the correct alternative:
The slope at any point of a curve y = f(x) is given by `("d"y)/("d"x) - 3x^2` and it passes through (-1, 1). Then the equation of the curve is
The general solution of the differential equation of all circles having centre at A(- 1, 2) 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 ______.
If y = (tan–1 x)2 then `(x^2 + 1)^2 (d^2y)/(dx^2) + 2x(x^2 + 1) (dy)/(dx)` = ______.
The differential equation of all parabolas whose axis is Y-axis, is ______.
The differential equation of the family of circles touching Y-axis at the origin is ______.
The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis is ______.
The differential equation for a2y = log x + b, is ______.
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.
The differential equation whose solution represents the family \[x^{2}y=4e^{x}+c\], where c is an arbitrary constant, is
