Advertisements
Advertisements
Question
Find the differential equation all parabolas having a length of latus rectum 4a and axis is parallel to the axis.
Advertisements
Solution
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
RELATED QUESTIONS
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:
y = e−2x (A cos x + B sin x)
Find the differential equation of the ellipse whose major axis is twice its minor axis.
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 = `(sin^-1 "x")^2 + "c"; (1 - "x"^2) ("d"^2"y")/"dx"^2 - "x" "dy"/"dx" = 2`
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:
`"dy"/"dx" = "e"^("x + y") + "x"^2 "e"^"y"`
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"`
Choose the correct option from the given alternatives:
The differential equation of y = `"c"^2 + "c"/"x"` is
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:
The solution of `"dy"/"dx" = ("y" + sqrt("x"^2 - "y"^2))/"x"` is
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 - a)2 = b(x + 4)
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" = "x"^2"y" + "y"`
Select and write the correct alternative from the given option for the question
Solution of the equation `x ("d"y)/("d"x)` = y log y is
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 y = (c1 + c2)ex
Find the differential equation of family of all ellipse whose major axis is twice the minor axis
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
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 equation of the family of all the parabolas with latus rectum 4a and whose axes are parallel to the x-axis
Find the differential equation of the family of parabolas with vertex at (0, –1) and having axis along the y-axis
Find the differential equation corresponding to the family of curves represented by the equation y = Ae8x + Be –8x, where A and B are arbitrary constants
The rate of disintegration of a radio active element at time t is proportional to its mass, at the time. Then the time during which the original mass of 1.5 gm. Will disintegrate into its mass of 0.5 gm. is proportional to ______.
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.
Solve the following differential equation:
`xsin(y/x)dy = [ysin(y/x) - x]dx`
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 the family of circles touching Y-axis at the origin is ______.
The differential equation of all circles passing through the origin and having their centres on the X-axis is ______.
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.
