Advertisements
Advertisements
प्रश्न
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = `sqrt("a" cos (log "x") + "b" sin (log "x"))`
Advertisements
उत्तर
y = `sqrt("a" cos (log "x") + "b" sin (log "x"))`
∴ y2 = a cos (log x) + b sin (log x) ....(1)
Differentiating both sides w.r.t. x, we get
`"2y" "dy"/"dx" = "a" "d"/"dx" [cos (log "x")] + "b" "d"/"dx" [sin (log "x")]`
`= "a" [ - sin (log "x")] * "d"/"dx" (log "x") + "b" cos (log "x") * "d"/"dx" (log "x")`
`= - "a" sin (log "x") xx 1/"x" + "b" cos (log "x") xx 1/"x"`
∴ `"2xy" "dy"/"dx" = - "a" sin (log "x") + "b" cos (log "x")`
Differentiating again w.r.t. x, we get
`2 ["xy" * "d"/"dx" ("dy"/"dx") + "dy"/"dx" * "d"/"dx" ("xy")]`
`= - "a" "d"/"dx" [sin (log "x")] + "b" "d"/"dx" [cos (log "x")]`
∴ `2 ["xy" ("d"^2"y")/"dx"^2 + "dy"/"dx" ("x" "dy"/"dx" + "y" xx 1)]`
`= - "a" cos (log "x") * "d"/"dx" (log "x") + "b"[- sin (log "x")] * "d"/"dx" (log "x")`
∴ `2"xy" ("d"^2"y")/"dx"^2 + 2"x" ("dy"/"dx")^2 + "2y" "dy"/"dx"
`= - "a" cos (log "x") xx 1/"x" - "b" sin (log "x") xx 1/"x"`
∴ `2"x"^2"y" ("d"^2"y")/"dx"^2 + 2"x"^2("dy"/"dx")^2 + 2"xy" "dy"/"dx"`
`= -["a" cos (log "x") + "b" sin (log "x")] = - "y"^2` ......[By (1)]
∴ `2"x"^2"y" ("d"^2"y")/"dx"^2 + 2"x"^2 ("dy"/"dx")^2 + 2"xy" "dy"/"dx" + "y"^2 = 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:
x3 + y3 = 4ax
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 = Ae5x + Be-5x
Find the differential equation all parabolas having a length of latus rectum 4a and axis is parallel to the axis.
In the following example verify that the given expression is a solution of the corresponding differential equation:
xy = log y +c; `"dy"/"dx" = "y"^2/(1 - "xy")`
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`
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`
Solve the following differential equation:
`log ("dy"/"dx") = 2"x" + 3"y"`
Solve the following differential equation:
`"sec"^2 "x" * "tan y" "dx" + "sec"^2 "y" * "tan x" "dy" = 0`
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:
(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:
The solution of `("x + y")^2 "dy"/"dx" = 1` is
The integrating factor of linear differential equation `x dy/dx + 2y = x^2 log x` is ______.
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
Solve the following differential equation:
`"dy"/"dx" + "y cot x" = "x"^2 "cot x" + "2x"`
Solve the following differential equation:
`"dx"/"dy" + "8x" = 5"e"^(- 3"y")`
Select and write the correct alternative from the given option for the question
General solution of `y - x ("d"y)/("d"x)` = 0 is
Find the differential equation of family of lines making equal intercepts on coordinate axes
Find the differential equation by eliminating arbitrary constants from the relation y = (c1 + c2x)ex
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
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.
Form the differential equation of all straight lines touching the circle x2 + y2 = r2
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 elimination of the arbitrary constant m from the equation y = emx gives the differential equation ______.
Solve the following differential equation:
`xsin(y/x)dy = [ysin(y/x) - x]dx`
The differential equation whose solution is (x – h)2 + (y – k)2 = a2 is (where a is a constant) ______.
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 circles passing through the origin and having their centres on the X-axis is ______.
The differential equation for a2y = log x + b, 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
Find the particular solution of the differential equation `x^2 dy/dx + y^2 = xy dy/dx`, if y = 1 when x = 1.
