Advertisements
Advertisements
प्रश्न
Solve the following differential equation:
`"dy"/"dx" = ("2y" - "x")/("2y + x")`
उत्तर
`"dy"/"dx" = ("2y" - "x")/("2y + x")` ....(1)
Put y = vx ∴ `"dy"/"dx" = "v + x""dv"/"dx"`
∴ (1) becomes, `"v + x""dv"/"dx" = ("2vx - x")/("2vx + x")`
∴ `"v + x""dv"/"dx" = ("2v" - 1)/("2v" + 1)`
∴ `"x""dv"/"dx" = ("2v" - 1)/("2v" + 1) - "v" = ("2v" - 1 - "2v"^2 - "v")/("2v + 1")`
∴ `"x""dv"/"dx" = - (("2v"^2 - "v" + 1)/("2v" + 1))`
∴ `("2v" + 1)/("2v"^2 - "v" + 1) "dv" = - 1/"x" "dx"`
Integrating both sides, we get
`int ("2v" + 1)/("2v"^2 - "v" + 1) "dv" = - int 1/"x" "dx"`
∴ `int (1/2 ("4v" - 1) + 3/2)/("2v"^2 - "v" + 1) "dv" = - int 1/"x" "dx"`
∴ `1/2 int ("4v" - 1)/("2v"^2 - "v" + 1) "dv" + 3/2 int 1/("2v"^2 - "v" + 1) "dv" = - int 1/"x"`
∴ `1/2 int ("4v" - 1)/("2v"^2 - "v" + 1) "dv" + 3/4 int 1/("v"^2 - 1/2"v" + 1/2)"dv" = - int 1/"x" "dx"`
∴ `1/2 int ("4v" - 1)/("2v"^2 - "v" + 1) "dv" + 3/4 int 1/(("v"^2 - 1/2"v" + 1/16) + 7/16) "dv" = - int 1/"x" "dx"`
∴ `1/2 int ("4v" - 1)/("2v"^2 - "v" + 1) "dv" + 3/4int 1/(("v" - 1/4)^2 + (sqrt7/4)^2)"dv" = - int 1/"x" "dx"`
∴ `1/2 log |2"v"^2 - "v" + 1| + 3/4 xx 1/((sqrt7/4)) tan^-1 |("v" - 1/4)/((sqrt7/4))| = - log |x| + "c"_1 .....[because "d"/"dv" (2"v"^2 - "v" + 1) = 4"v" - 1 and int ("f"'("v"))/("f"("v")) "dv" = log |"f"("v")| + c]`
∴ `1/2 log |2 ("y"^2/"x"^2) - "y"/"x" + 1| + 3/sqrt7 tan^-1 ((4("y"/"x") - 1)/sqrt7) = - log |"x"| + "c"_1`
∴ `1/2 log |(2"y"^2 - "xy" + "x"^2)/"x"^2| + 3/sqrt7 tan^-1 ((4"y - x")/(sqrt7"x")) = - log |"x"| + "c"_1`
∴ `log |("x"^2 - "xy" + "2y"^2)/"x"^2| + 6/sqrt7 tan^-1 (("4y - x")/(sqrt7"x")) = - 2 log |"x"| + 2"c"_1`
∴ `log |"x"^2 - "xy" + "2y"^2| - log"x"^2 + 6/sqrt7 tan^-1 (("4y - x")/(sqrt7"x")) = - log "x"^2 + "c"_1 "where" "c" = 2"c"_1`
∴ `log |"x"^2 - "xy" + "2y"^2| + 6/sqrt7 tan^-1 (("4y - x")/(sqrt7"x")) = "c"`
This is the general solution.
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:
y2 = (x + c)3
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:
c1x3 + c2y2 = 5
Find the differential equation all parabolas having a length of latus rectum 4a and axis is parallel to the axis.
Form the differential equation of family of lines parallel to the line 2x + 3y + 4 = 0.
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`
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:
`"y" - "x" "dy"/"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:
`cos("dy"/"dx") = "a", "a" ∈ "R", "y"(0) = 2`
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:
`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:
x2 + y2 = a2 is a solution of
Choose the correct option from the given alternatives:
The solution of the differential equation `"dy"/"dx" = sec "x" - "y" tan "x"`
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.
`"y" = 3 "cos" (log "x") + 4 sin (log "x"); "x"^2 ("d"^2"y")/"dx"^2 + "x" "dy"/"dx" + "y" = 0`
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)`
Solve the following differential equation:
x dy = (x + y + 1) dx
Solve the following differential equation:
y log y = (log y2 - x) `"dy"/"dx"`
Find the particular solution of the following differential equation:
`"dy"/"dx" - 3"y" cot "x" = sin "2x"`, when `"y"(pi/2) = 2`
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 by eliminating arbitrary constants from the relation x2 + y2 = 2ax
Find the differential equation from the relation x2 + 4y2 = 4b2
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 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 = ______.
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 ______.
Find the particular solution of the differential equation `x^2 dy/dx + y^2 = xy dy/dx`, if y = 1 when x = 1.
