Advertisements
Advertisements
Question
Find the particular solution of the following differential equation:
(x + y)dy + (x - y)dx = 0; when x = 1 = y
Advertisements
Solution
(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
RELATED QUESTIONS
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 = c1e2x + c2e5x
Find the differential equation all parabolas having a length of latus rectum 4a and axis is parallel to the axis.
Find the differential equation of the ellipse whose major axis is twice its minor 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:
xy = log y +c; `"dy"/"dx" = "y"^2/(1 - "xy")`
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 = xm; `"x"^2 ("d"^2"y")/"dx"^2 - "mx" "dy"/"dx" + "my" = 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:
`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:
`"dy"/"dx" = "e"^("x + y") + "x"^2 "e"^"y"`
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
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"`
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.
`"y" = 3 "cos" (log "x") + 4 sin (log "x"); "x"^2 ("d"^2"y")/"dx"^2 + "x" "dy"/"dx" + "y" = 0`
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = a sin (x + b)
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.
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" = "x"^2"y" + "y"`
Solve the following differential equation:
x dy = (x + y + 1) dx
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
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 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
Find the differential equation by eliminating arbitrary constants from the relation x2 + y2 = 2ax
Find the differential equation by eliminating arbitrary constants from the relation y = (c1 + c2x)ex
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
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 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.
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
