Advertisements
Advertisements
प्रश्न
Find the particular solution of the following differential equation:
(x + y)dy + (x - y)dx = 0; when x = 1 = y
Advertisements
उत्तर
(x + y)dy + (x - y)dx = 0
∴ (x + y)dy = - (x - y)dx
∴ `"dy"/"dx" = ("y - x")/("y + x")` ...(1)
Put y = vx ∴ `"dy"/"dx" = "v" + "x" "dv"/"dx"`
∴ (1) becomes, `"v" + "x" "dv"/"dx" = ("vx" - "x")/("vx" + "x") = ("v - 1")/("v + 1")`
∴ `"x" "dv"/"dx" = ("v - 1")/("v + 1") - "v" = ("v" - 1 - "v"^2 - "v")/("v + 1")`
∴ `"x" "dv"/"dx" = - ((1 + "v"^2)/(1 + "v"))`
∴ `(1 + "v")/(1 + "v"^2) "dv" = - 1/"x" "dx"`
Integrating both sides, we get
`int (1 + "v")/(1 + "v"^2) "dv" = - int 1/"x" "dx"`
∴ `int (1/(1 + "v"^2) + "v"/(1 + "v"^2))"dv" = - int 1/"x" "dx"`
∴ `int 1/(1 + "v"^2) "dv" + 1/2 int "2v"/(1 + "v"^2)"dv" = - int 1/"x" "dx"`
∴ `tan^-1 "v" + 1/2 log |1 + "v"^2| = - log "x" + "c"` .....`[because "d"/"dv" (1 + "v"^2) = 2"v" and int ("f"'(x))/("f"("x")) "dv" = log |"f"("v")| + "c"]`
∴ `tan^-1 ("y"/"x") + 1/2 log |1 + "y"^2/"x"^2| = log |"x"| + "c"`
∴ `tan^-1 ("y"/"x") + 1/2 log |("x"^2 + "y"^2)/"x"^2| = - log |"x"| + "c"`
∴ `tan^-1 ("y"/"x") + 1/2 log "x"^2 + "y"^2 - 1/2 log |"x"^2| = - log |"x"| + "c"`
∴ `tan^-1 ("y"/"x") + log sqrt("x"^2 + "y"^2) - log |"x"| = - log |"x"| + "c"`
∴ `tan^-1 ("y"/"x") + log sqrt("x"^2 + "y"^2) = "c"`
This is the general solution.
When x = 1 = y, we have
`tan^-1 (1) + log sqrt(1^2 + 1^2) = "c"`
∴ `tan^-1 (tan pi/4) + log sqrt 2 = "c"`
∴ c = `pi/4 + log sqrt2`
∴ the particular solution is
∴ `tan^-1 ("y"/"x") + log sqrt("x"^2 + "y"^2) = pi/4 + log sqrt2`
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:
y = A cos (log x) + B sin (log x)
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:
y = c1e2x + c2e5x
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
c1x3 + c2y2 = 5
Form the differential equation of family of lines parallel to the line 2x + 3y + 4 = 0.
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 = `(sin^-1 "x")^2 + "c"; (1 - "x"^2) ("d"^2"y")/"dx"^2 - "x" "dy"/"dx" = 2`
Solve the following differential equation:
`log ("dy"/"dx") = 2"x" + 3"y"`
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
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:
`"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:
The differential equation of y = `"c"^2 + "c"/"x"` is
The particular solution of `dy/dx = xe^(y - x)`, when x = y = 0 is ______.
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
`"y"^2 = "a"("b - x")("b + x")`
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.
`"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 parabolas which have 4b as latus rectum and whose axis is parallel to the Y-axis.
Solve the following differential equation:
x dy = (x + y + 1) dx
Find the particular solution of the following differential equation:
`("x + 2y"^2) "dy"/"dx" = "y",` when x = 2, y = 1
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
The general solution of `(dy)/(dx)` = e−x 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
Form the differential equation of family of standard circle
Form the differential equation of y = (c1 + c2)ex
Find the differential equation by eliminating arbitrary constants from the relation x2 + y2 = 2ax
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.
Find the differential equation of the family of all non-vertical lines in a plane
Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin
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 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 representing the family of parabolas having vertex at origin and axis along positive direction of X-axis is ______.
Form the differential equation of all lines which makes intercept 3 on x-axis.
The differential equation of all parabolas whose axis is Y-axis, is ______.
The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis 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.
