Advertisements
Advertisements
Question
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = `sqrt("a" cos (log "x") + "b" sin (log "x"))`
Advertisements
Solution
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
RELATED QUESTIONS
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 = e−2x (A cos x + B sin x)
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`
Solve the following differential equation:
`"sec"^2 "x" * "tan y" "dx" + "sec"^2 "y" * "tan x" "dy" = 0`
Solve the following differential equation:
cos x . cos y dy − sin x . sin y dx = 0
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:
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
For the following differential equation find the particular solution satisfying the given condition:
`("x" + 1) "dy"/"dx" - 1 = 2"e"^-"y" , "y" = 0`, when x = 1
Reduce the following differential equation to the variable separable form and hence solve:
`("x - y")^2 "dy"/"dx" = "a"^2`
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.
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
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.
`"xy" = "ae"^"x" + "be"^-"x" + "x"^2; "x" ("d"^2"y")/"dx"^2 + 2 "dy"/"dx" + "x"^2 = "xy" + 2`
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 the lines which are normal to the line 3x + 2y + 7 = 0.
Solve the following differential equation:
`"dy"/"dx" = "x"^2"y" + "y"`
Solve the following differential equation:
x dy = (x + y + 1) dx
Find the particular solution of the following differential equation:
`"dy"/"dx" - 3"y" cot "x" = sin "2x"`, when `"y"(pi/2) = 2`
Find the particular solution of the following differential equation:
y(1 + log x) = (log xx) `"dy"/"dx"`, when y(e) = e2
Find the particular solution of the following differential equation:
`2e ^(x/y) dx + (y - 2xe^(x/y)) dy = 0," When" y (0) = 1`
The general solution of `(dy)/(dx)` = e−x is ______.
Select and write the correct alternative from the given option for the question
The solution of `("d"y)/("d"x)` = 1 is
Find the differential equation of family of lines making equal intercepts on coordinate axes
Form the differential equation of family of standard circle
Find the differential equation by eliminating arbitrary constants from the relation x2 + y2 = 2ax
Find the differential equation by eliminating arbitrary constants from the relation y = (c1 + c2x)ex
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
Find the differential equation of the family of parabolas with vertex at (0, –1) and having axis along the y-axis
If `x^2 y^2 = sin^-1 sqrt(x^2 + y^2) + cos^-1 sqrt(x^2 + y^2)`, then `"dy"/"dx"` = ?
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 ______.
If m and n are respectively the order and degree of the differential equation of the family of parabolas with focus at the origin and X-axis as its axis, then mn - m + n = ______.
The differential equation for all the straight lines which are at the distance of 2 units from the origin is ______.
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.
Form the differential equation of all concentric circles having centre at the origin.
The differential equation whose solution represents the family \[x^{2}y=4e^{x}+c\], where c is an arbitrary constant, is
