Advertisements
Advertisements
Question
Solve the following differential equation:
(x2 + y2)dx - 2xy dy = 0
Advertisements
Solution
(x2 + y2)dx - 2xy dy = 0
∴ 2xy dy = (x2 + y2)dx
∴ `"dy"/"dx" = ("x"^2 + "y"^2)/"2xy"` ....(1)
Put y = vx
∴ `"dy"/"dx" = "v"+ ("xdv")/"dx"`
∴ (1) becomes, v + x`"dv"/"dx" = ("x"^2 + "v"^2"x"^2)/("2x"("vx"))`
∴ `"v + x""dv"/"dx" = (1 + "v"^2)/"2v"`
∴ `"x""dv"/"dx" = (1 + "v"^2)/"2v" - "v" = (1 + "v"^2 - 2"v"^2)/"2v"`
∴ `"x""dv"/"dx" = (1 - "v"^2)/"2v"`
∴ `"2v"/(1 - "v"^2)"dv" = 1/"x" "dx"`
Integrating both sides, we get
`int"2v"/(1 - "v"^2)"dv" = int 1/"x" "dx"`
`- int"- 2v"/(1 - "v"^2)"dv" = int 1/"x" "dx"`
∴ - log |1 - v2| = log x + log c1 ....`[because "d"/"dv" (1 - "v"^2) = - 2"v" and int("f"'("x"))/("f"("x")) "dx" = log |"f"("x")| + "c"]`
∴ `log |1/(1 - "v"^2)| = log "c"_1 "x"`
∴ `log |1/(1 - ("y"^2/"x"^2))| = log "c"_1 "x"`
∴ `log |"x"^2/("x"^2 - "y"^2)| = log "c"_1 "x"`
∴ `"x"^2/("x"^2 - "y"^2) = "c"_1"x"`
∴ `"x"^2 - "y"^2 = 1/"c"_1 "x"`
∴ `"x"^2 - "y"^2 = "cx"`, where c = `1/"c"_1`
This is the general solution.
APPEARS IN
RELATED QUESTIONS
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:
y = c1e2x + c2e5x
Find the differential equation of all circles having radius 9 and centre at point (h, k).
Solve the following differential equation:
`"y" - "x" "dy"/"dx" = 0`
Solve the following differential equation:
`"sec"^2 "x" * "tan y" "dx" + "sec"^2 "y" * "tan x" "dy" = 0`
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:
`("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:
(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:
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"`
Choose the correct option from the given alternatives:
`"x"^2/"a"^2 - "y"^2/"b"^2 = 1` is a solution of
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 - a)2 = b(x + 4)
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:
`"dy"/"dx" - 3"y" cot "x" = sin "2x"`, when `"y"(pi/2) = 2`
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
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
Find the differential equation of family of lines making equal intercepts on coordinate axes
Form the differential equation of y = (c1 + c2)ex
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
Form the differential equation of all straight lines touching the circle x2 + y2 = r2
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 of the family of all the parabolas with latus rectum 4a and whose axes are parallel to the x-axis
Find the differential equation of the family of parabolas with vertex at (0, –1) and having axis along the y-axis
The differential equation for all the straight lines which are at the distance of 2 units from the origin is ______.
Form the differential equation of all lines which makes intercept 3 on x-axis.
Solve the following differential equation:
`xsin(y/x)dy = [ysin(y/x) - x]dx`
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 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 ______.
The differential equation for a2y = log x + b, is ______.
If 2x = `y^(1/m) + y^(-1/m)`, then show that `(x^2 - 1) (dy/dx)^2` = m2y2
Form the differential equation of all concentric circles having centre at the origin.
The differential equation whose solution represents the family \[x^{2}y=4e^{x}+c\], where c is an arbitrary constant, is
