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:
Ax2 + By2 = 1
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y2 = (x + c)3
Find the differential equation of the ellipse whose major axis is twice its minor axis.
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`
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:
`"dy"/"dx" = - "k",` where k is a constant.
Solve the following differential equation:
`"y"^3 - "dy"/"dx" = "x"^2 "dy"/"dx"`
Solve the following differential equation:
`"dy"/"dx" = "e"^("x + y") + "x"^2 "e"^"y"`
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:
`("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:
`cos^2 ("x - 2y") = 1 - 2 "dy"/"dx"`
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
Choose the correct option from the given alternatives:
The solution of `("x + y")^2 "dy"/"dx" = 1` is
Choose the correct option from the given alternatives:
The solution of `"dy"/"dx" + "y" = cos "x" - sin "x"`
Choose the correct option from the given alternatives:
The solution of the differential equation `"dy"/"dx" = sec "x" - "y" tan "x"`
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:
y(1 + log x) = (log xx) `"dy"/"dx"`, when y(e) = e2
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 all ellipse whose major axis is twice the minor axis
Find the differential equation from the relation x2 + 4y2 = 4b2
Find the differential equation of the family of all non-horizontal lines in a plane
Find the differential equation of the family of parabolas with vertex at (0, –1) and having axis along the y-axis
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 ______.
If `x^2 y^2 = sin^-1 sqrt(x^2 + y^2) + cos^-1 sqrt(x^2 + y^2)`, then `"dy"/"dx"` = ?
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 ______.
The differential equation whose solution is (x – h)2 + (y – k)2 = a2 is (where a is a constant) ______.
The differential equation of all parabolas whose axis is Y-axis, is ______.
The differential equation of all circles passing through the origin and having their centres on the X-axis is ______.
The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis is ______.
Solve the differential equation
ex tan y dx + (1 + ex) sec2 y dy = 0
