Advertisements
Advertisements
प्रश्न
Solve the following differential equation:
`x ("d"y)/("d"x) = y - xcos^2(y/x)`
Advertisements
उत्तर
Given `x ("d"y)/("d"x) = y - x cos^2 y/x`
The equation can be written as
`("d"y)/("d"x) = (y - cos^2 y/x)/x` ..........(1)
This is a homogeneous differential equation.
y = vx
`("d"y)/("d"x) = "v"(1) + x "dv"/("d"x)`
Substituting `("d"y)/("d"x)` value in equation (1), we get
`"v" + (x"dv")/("d"x) = ("v"x - x cos^2 ((vx)/x))/x`
`"v" + (x"dv")/("d"x) = ("v"x - x cos^2("v"))/x`
`"v" + (x"dv")/("d"x) = x (("v" - cos^2"v"))/x`
`x "dv"/("d"x) = "v" - cos^2"v" - "v"`
`"dv"/("d"x) = (- cos^2"v")/x`
`"dv"/(cos^2"v") = (-"d"x)/x`
Integrating on both sides, we get
`int sec^2"v" "d"x = - int ("d"x)/x`
tan v = – log x + log C
tan v = log C – log x
tan v = `log ("C"/x)`
etan v = `"C"/x`
C = xetan v
C = `xe"^(tan y/x)` is a required equation.
APPEARS IN
संबंधित प्रश्न
The velocity v, of a parachute falling vertically satisfies the equation `"v" (dv)/(dx) = "g"(1 - v^2/k^2)` where g and k are constants. If v and are both initially zero, find v in terms of x
Solve the following differential equation:
`y"d"x + (1 + x^2)tan^-1x "d"y`= 0
Solve the following differential equation:
`sin ("d"y)/("d"x)` = a, y(0) = 1
Solve the following differential equation:
`tan y ("d"y)/("d"x) = cos(x + y) + cos(x - y)`
Solve the following differential equation:
`[x + y cos(y/x)] "d"x = x cos(y/x) "d"y`
Choose the correct alternative:
The general solution of the differential equation `log(("d"y)/("d"x)) = x + y` is
Choose the correct alternative:
The number of arbitrary constants in the general solutions of order n and n +1are respectively
Solve : cos x(1 + cosy) dx – sin y(1 + sinx) dy = 0
Solve: `("d"y)/("d"x) = y sin 2x`
Solve the following homogeneous differential equation:
`x ("d"y)/("d"x) - y = sqrt(x^2 + y^2)`
Solve the following homogeneous differential equation:
`("d"y)/("d"x) = (3x - 2y)/(2x - 3y)`
Solve the following homogeneous differential equation:
(y2 – 2xy) dx = (x2 – 2xy) dy
Solve the following homogeneous differential equation:
An electric manufacturing company makes small household switches. The company estimates the marginal revenue function for these switches to be (x2 + y2) dy = xy dx where x represents the number of units (in thousands). What is the total revenue function?
Solve the following:
`("d"y)/("d"x) - y/x = x`
Solve the following:
If `("d"y)/("d"x) + 2 y tan x = sin x` and if y = 0 when x = `pi/3` express y in term of x
Choose the correct alternative:
If y = ex + c – c3 then its differential equation is
Choose the correct alternative:
A homogeneous differential equation of the form `("d"y)/("d"x) = f(y/x)` can be solved by making substitution
Choose the correct alternative:
Which of the following is the homogeneous differential equation?
Solve (D2 – 3D + 2)y = e4x given y = 0 when x = 0 and x = 1
