Advertisements
Advertisements
प्रश्न
Solve the following differential equation:
`[x + y cos(y/x)] "d"x = x cos(y/x) "d"y`
Advertisements
उत्तर
The given equation can be written as
`("d"y)/("d"x) = (x + y cos(y/x))/(x cos y/x)` ........(1)
This is a homogeneous differential equations.
Put y = vx
`("d"y)/("d"x) = "v" + x "dv"/("d"x)`
1 ⇒ ∴ `"v" + x "dv"/("d"x) = (x + "v"x cos((vx)/x))/(x cos((vx)/x))`
`"v" + x "dv"/("d"x) = (x[1 + "v" cos "v"])/cos "v"`
`x "dv"/("d"x) = (1 + "v" cos "v")/cos"v" - "v"`
`x "dv"/("d"x) = 1/cos"v"`
cos v dv = `("d"x)/x`
On integration we obtain
`int cos"v" "d"v = int ("d"x)/x`
sin v = log x + log c
`sin(y/x)` = log x + log c
`sin(y/x)` = log |cx|
Which gives the required solution.
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:
x cos y dy = ex(x log x + 1) dx
Solve the following differential equation:
`("d"y)/("d"x) = tan^2(x + y)`
Solve the following differential equation:
`(y^2 - 2xy) "d"x = (x^2 - 2xy) "d"y`
Choose the correct alternative:
The general solution of the differential equation `log(("d"y)/("d"x)) = x + y` is
Solve: ydx – xdy = 0 dy
Solve: (1 – x) dy – (1 + y) dx = 0
Solve the following homogeneous differential equation:
`(x - y) ("d"y)/("d"x) = x + 3y`
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:
`("d"y)/("d"x) - y/x = x`
Choose the correct alternative:
If y = ex + c – c3 then its differential equation is
Choose the correct alternative:
The solution of the differential equation `("d"y)/("d"x) + "P"y` = Q where P and Q are the function of x is
Choose the correct alternative:
The differential equation of x2 + y2 = a2
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:
The variable separable form of `("d"y)/("d"x) = (y(x - y))/(x(x + y))` by taking y = vx and `("d"y)/("d"x) = "v" + x "dv"/("d"x)` is
Choose the correct alternative:
Which of the following is the homogeneous differential equation?
Form the differential equation having for its general solution y = ax2 + bx
Solve (D2 – 3D + 2)y = e4x given y = 0 when x = 0 and x = 1
