Advertisements
Advertisements
प्रश्न
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
c1x3 + c2y2 = 5
उत्तर
c1x3 + c2y2 = 5 .....(1)
Differentiating twice w.r.t. x, we get
`"c"_1 xx "3x"^2 + "c"_2 xx "2y" "dy"/"dx" = 0`
∴ `3"c"_1"x"^2 + 2"c"_2"y" "dy"/"dx" = 0` ....(2)
Differentiating again w.r.t. x, we get
`3"c"_1 xx "2x" + 2"c"_2 ["y"."d"/"dx"("dy"/"dx") + "dy"/"dx" * "dy"/"dx"] = 0`
∴ `6"c"_1"x" + 2"c"_2 ["y" ("d"^2"y")/"dx"^2 + ("dy"/"dx")^2] = 0`
The equations (1), (2) and (3) in c1, c2 are consistent.
∴ determinant of their consistency is zero.
∴ `|("x"^3, "y"^2, 5),("3x"^2, "2y""dy"/"dx", 0),(6"x", "2y" ("d"^2"y")/"dx"^2 + 2("dy"/"dx")^2, 0)|`
∴ `"x"^3 (0 - 0) - "y"^2(0 - 0) + 5["6x"^2"y" ("d"^2"y")/"dx"^2 + "6x"^2("dy"/"dx")^2 - 12"xy" "dy"/"dx"] = 0`
∴ `"6x"^2"y" ("d"^2"y")/"dx"^2 + "6x"^2("dy"/"dx")^2 - 12"xy" "dy"/"dx" = 0`
∴ `"xy" ("d"^2"y")/"dx"^2 + "x"("dy"/"dx")^2 - "2y" "dy"/"dx" = 0`
This is the required D.E.
Notes
The answer in the textbook is incorrect.
APPEARS IN
संबंधित प्रश्न
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
Ax2 + By2 = 1
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 = a + `"a"/"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.
Find the differential equation of all circles having radius 9 and centre at point (h, k).
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 = `"e"^"ax"; "x" "dy"/"dx" = "y" log "y"`
Solve the following differential equation:
`log ("dy"/"dx") = 2"x" + 3"y"`
Solve the following differential equation:
`"dy"/"dx" = - "k",` where k is a constant.
Solve the following differential equation:
`"y"^3 - "dy"/"dx" = "x"^2 "dy"/"dx"`
For the following differential equation find the particular solution satisfying the given condition:
`cos("dy"/"dx") = "a", "a" ∈ "R", "y"(0) = 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 solution of `"dy"/"dx" = ("y" + sqrt("x"^2 - "y"^2))/"x"` is
Choose the correct option from the given alternatives:
The solution of `"dy"/"dx" + "y" = cos "x" - sin "x"`
The integrating factor of linear differential equation `x dy/dx + 2y = x^2 log x` is ______.
The particular solution of `dy/dx = xe^(y - x)`, when x = y = 0 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 + "y"^2 = "r"^2; "x" "dy"/"dx" + "r" sqrt(1 + ("dy"/"dx")^2) = "y"`
In the following example verify that the given function is a solution of the differential equation.
`"xy" = "ae"^"x" + "be"^-"x" + "x"^2; "x" ("d"^2"y")/"dx"^2 + 2 "dy"/"dx" + "x"^2 = "xy" + 2`
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:
x dy = (x + y + 1) dx
Find the particular solution of the following differential equation:
y(1 + log x) = (log xx) `"dy"/"dx"`, when y(e) = e2
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
Find the differential equation of family of lines making equal intercepts on coordinate axes
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
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
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 differential equation of all lines perpendicular to the line 5x + 2y + 7 = 0 is ____________.
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 ______.
Solve the differential equation
cos2(x – 2y) = `1 - 2dy/dx`
If 2x = `y^(1/m) + y^(-1/m)`, then show that `(x^2 - 1) (dy/dx)^2` = m2y2
