Advertisements
Advertisements
प्रश्न
Solve the following differential equation:
(x2 + y2)dx - 2xy dy = 0
Advertisements
उत्तर
(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
संबंधित प्रश्न
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
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:
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 = 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 = `"e"^"ax"; "x" "dy"/"dx" = "y" log "y"`
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:
cos x . cos y dy − sin x . sin y 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:
`("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:
`"x + y""dy"/"dx" = sec("x"^2 + "y"^2)`
Choose the correct option from the given alternatives:
The differential equation of all circles having their centres on the line y = 5 and touching the X-axis 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"`
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.
`"x"^2 = "2y"^2 log "y", "x"^2 + "y"^2 = "xy" "dx"/"dy"`
Form the differential equation of all the lines which are normal to the line 3x + 2y + 7 = 0.
Solve the following differential equation:
`"dy"/"dx" + "y cot x" = "x"^2 "cot x" + "2x"`
Find the particular solution of the following differential equation:
`("x + 2y"^2) "dy"/"dx" = "y",` when x = 2, y = 1
Form the differential equation of y = (c1 + c2)ex
Find the differential equation by eliminating arbitrary constants from the relation y = (c1 + c2x)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
Find the differential equation of the family of all non-vertical lines in a plane
Find the differential equation of the family of all non-horizontal 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 parabolas with vertex at (0, –1) and having axis along the y-axis
Find the differential equation of the curve represented by xy = aex + be–x + x2
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 = ______.
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 ______.
If y = (tan–1 x)2 then `(x^2 + 1)^2 (d^2y)/(dx^2) + 2x(x^2 + 1) (dy)/(dx)` = ______.
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`
If 2x = `y^(1/m) + y^(-1/m)`, then show that `(x^2 - 1) (dy/dx)^2` = m2y2
Solve the differential equation
ex tan y dx + (1 + ex) sec2 y dy = 0
Form the differential equation of all concentric circles having centre at the origin.
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.
