Advertisements
Advertisements
प्रश्न
Find the particular solution of the following differential equation:
`2e ^(x/y) dx + (y - 2xe^(x/y)) dy = 0," When" y (0) = 1`
Advertisements
उत्तर
`(1 + 2"e"^("x"//"y"))"dx" + 2"e"^("x"//"y")(1 - "x"/"y")"dy" = 0`
∴ `(1 + 2"e"^("x"//"y"))"dx" = - 2"e"^("x"//"y")(1 - "x"/"y")"dy"`
∴ `(1 + 2"e"^("x"//"y"))"dx" = 2"e"^("x"//"y")("x"/"y" - 1)"dy"`
∴ `"dy"/"dx" = (2"e"^("x"//"y")("x"/"y" - 1))/(1 + 2"e"^("x"//"y"))` .....(1)
Put x = vy
∴ `"dx"/"dy" = "v" + "y" "dv"/"dy"`
∴ (1) becomes, `"v" + "y" "dv"/"dy" = (2"e"^"v"("v - 1"))/(1 + "2e"^"v")`
∴ `"y" "dv"/"dy" = (2"e"^"v"("v - 1"))/(1 + "2e"^"v") - "v"`
`= (2"ve"^"v" - 2"e"^"v" - "v" - 2"ve"^"v")/(1 + "2e"^"v")`
`= - (("v" + 2"e"^"v")/(1 + "2e"^"v"))`
∴ `((1 + 2"e"^"v")/("v" + 2"e"^"v"))"dv" ≡ - 1/"y" "dy"`
Integrating both sides, we get
`int ((1 + 2"e"^"v")/("v" + 2"e"^"v"))"dv" ≡ - int 1/"y" "dy"`
∴ log |v + 2ev| = - log y + log c ....`[because "d"/"dx" ("v" + "2e"^"v") = 1 + 2"e"^"v" and int("f"'("v"))/("f"("v")) "dv" = log |"f"("v")| + "c"]`
∴ log |v + 2ev| + log y = log c
∴ log |y (v + 2ev)| = log c
∴ y(v + 2ev) = c
∴ `"y"("x"/"y" + 2"e"^("x"//"y"))`= c
∴ x + 2yex/y = c
This is the general solution.
Now, y(0) = 1, i.e. when x = 0, y = 1
∴ 0 + 2(1)e0 = c
∴ c = 2
∴ the particular solution is x + 2yex/y = 2
Notes
The question is modified.
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 = Ae5x + Be-5x
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
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 all circles having radius 9 and centre at point (h, k).
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`
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
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")^2 "dy"/"dx" = "a"^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:
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
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.
`"y" = "e"^"ax" sin "bx"; ("d"^2"y")/"dx"^2 - 2"a" "dy"/"dx" + ("a"^2 + "b"^2)"y" = 0`
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`
Form the differential equation of all parabolas which have 4b as latus rectum and whose axis is parallel to the Y-axis.
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:
`"dy"/"dx" + "y cot x" = "x"^2 "cot x" + "2x"`
Select and write the correct alternative from the given option for the question
General solution of `y - x ("d"y)/("d"x)` = 0 is
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
Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0
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 all non-vertical lines in a plane
Find the differential equation of the family of circles passing through the origin and having their centres on the x-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
Find the differential equation of the curve represented by xy = aex + be–x + x2
The general solution of the differential equation of all circles having centre at A(- 1, 2) is ______.
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 representing the family of ellipse having foci either on the x-axis or on the y-axis centre at the origin and passing through the point (0, 3) 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 ______.
If 2x = `y^(1/m) + y^(-1/m)`, then show that `(x^2 - 1) (dy/dx)^2` = m2y2
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.
Solve the differential equation
ex tan y dx + (1 + ex) sec2 y dy = 0
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.
