Advertisements
Advertisements
प्रश्न
Solve the following differential equation:
`"dy"/"dx" = ("2y" - "x")/("2y + x")`
Advertisements
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:
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:
c1x3 + c2y2 = 5
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 = `"a" + "b"/"x"; "x" ("d"^2"y")/"dx"^2 + 2 "dy"/"dx" = 0`
Solve the following differential equation:
`(cos^2y)/x dy + (cos^2x)/y dx` = 0
Solve the following differential equation:
`2"e"^("x + 2y") "dx" - 3"dy" = 0`
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
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:
`"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.
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 `"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 ______.
Choose the correct option from the given alternatives:
The solution of the differential equation `"dy"/"dx" = sec "x" - "y" tan "x"`
The particular solution of `dy/dx = xe^(y - x)`, when x = y = 0 is ______.
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.
`"x"^2 = "2y"^2 log "y", "x"^2 + "y"^2 = "xy" "dx"/"dy"`
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:
`"dy"/"dx" = "x"^2"y" + "y"`
Solve the following differential equation:
x dy = (x + y + 1) dx
Solve the following differential equation:
y log y = (log y2 - x) `"dy"/"dx"`
Solve the following differential equation:
`"dx"/"dy" + "8x" = 5"e"^(- 3"y")`
Find the particular solution of the following differential equation:
`2e ^(x/y) dx + (y - 2xe^(x/y)) dy = 0," When" y (0) = 1`
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
Find the general solution of `("d"y)/("d"x) = (1 + y^2)/(1 + x^2)`
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
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 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 of all lines perpendicular to the line 5x + 2y + 7 = 0 is ____________.
The differential equation representing the family of parabolas having vertex at origin and axis along positive direction of X-axis is ______.
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 ______.
The differential equation of all parabolas whose axis is Y-axis, is ______.
The differential equation of the family of circles touching Y-axis at the origin is ______.
Solve the differential equation
cos2(x – 2y) = `1 - 2dy/dx`
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.
A particle is moving along the X-axis. Its acceleration at time t is proportional to its velocity at that time. Find the differential equation of the motion of the particle.
