Advertisements
Advertisements
प्रश्न
Choose the correct option from the given alternatives:
The solution of `"dy"/"dx" = ("y" + sqrt("x"^2 - "y"^2))/"x"` is
विकल्प
`sin^-1 ("y"/"x") = 2 log |"x"| + "c"`
`sin^-1 ("y"/"x") = log |"x"| + "c"`
`sin ("y"/"x") = log |"x"| + "c"`
`sin ("y"/"x") = 2 log |"x"| + "c"`
Advertisements
उत्तर
`sin^-1 ("y"/"x") = log |"x"| + "c"`
Hint:
`"dy"/"dx" = ("y" + sqrt("x"^2 - "y"^2))/"x"`
Put y = vx ∴ `"dy"/"dx" = "v + x" "dv"/"dx"`
∴ `"v + x" "dv"/"dx" = ("vx" + sqrt("x"^2 - "v"^2"x"^2))/"x" = "v" + sqrt(1 - "v"^2)`
∴ `"x" "dv"/"dx" = sqrt(1 - "v"^2)`
∴ `int 1/sqrt(1 - "v"^2) "dv" = int 1/"x" "dx"`
∴ sin-1 v = log |x| + c
∴ sin-1 v `("y"/"x") = log |x| + "c"`.
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 - a)2 = 4(x - b)
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:
y = e−2x (A cos x + B sin x)
Find the differential equation of the ellipse whose major axis is twice its minor axis.
Solve the following differential equation:
`log ("dy"/"dx") = 2"x" + 3"y"`
Solve the following differential equation:
`"dy"/"dx" = - "k",` where k is a constant.
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
For the following differential equation find the particular solution satisfying the given condition:
`cos("dy"/"dx") = "a", "a" ∈ "R", "y"(0) = 2`
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:
`cos^2 ("x - 2y") = 1 - 2 "dy"/"dx"`
Choose the correct option from the given alternatives:
x2 + y2 = a2 is a solution of
The integrating factor of linear differential equation `x dy/dx + 2y = x^2 log x` is ______.
The particular solution of `dy/dx = xe^(y - x)`, when x = y = 0 is ______.
Choose the correct option from the given alternatives:
`"x"^2/"a"^2 - "y"^2/"b"^2 = 1` is a solution of
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
(y - a)2 = b(x + 4)
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = `sqrt("a" cos (log "x") + "b" sin (log "x"))`
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:
`"dy"/"dx" = ("2y" - "x")/("2y + x")`
Solve the following differential equation:
`"dx"/"dy" + "8x" = 5"e"^(- 3"y")`
Find the particular solution of the following differential equation:
y(1 + log x) = (log xx) `"dy"/"dx"`, when y(e) = e2
Find the differential equation of family of lines making equal intercepts on coordinate axes
Find the differential equation of family of all ellipse whose major axis is twice the minor axis
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-horizontal lines in a plane
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
Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin
Choose the correct alternative:
The slope at any point of a curve y = f(x) is given by `("d"y)/("d"x) - 3x^2` and it passes through (-1, 1). Then the equation of the curve is
The general solution of the differential equation of all circles having centre at A(- 1, 2) is ______.
The elimination of the arbitrary constant m from the equation y = emx gives the differential equation ______.
The differential equation whose solution is (x – h)2 + (y – k)2 = a2 is (where a is a constant) ______.
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 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
cos2(x – 2y) = `1 - 2dy/dx`
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.
The differential equation whose solution represents the family \[x^{2}y=4e^{x}+c\], where c is an arbitrary constant, is
