Advertisements
Advertisements
प्रश्न
Solve the following differential equation:
`2xy"d"x + (x^2 + 2y^2)"d"y` = 0
Advertisements
उत्तर
The given differential equation can be written as
`("d"y)/("d"x) = (- 2xy)/(x^2 + 2y^2)` .........(1)
This is a homogenous differential equation
Putting y = vx
`("d"y)/("d"x) = "v"(1) + x "dv"/("d"x)`
`("d"y)/("d"x) = "v" + x "dv"/("d"x)`
(1) ⇒ `"v" + x "dv"/("d"x) = (-2x "v"x)/(x^2 + 2("v"x)^2)`
= `(- 2"v"x^2)/(x^2 + 2"v"^2x^2)`
= `(- x^2 2"v")/(x^2[1 + 2"v"^2])`
= `(- 2"v")/(1 + 2"v"^2)`
`"v" + x "dv"/("d"x) = (- 2"v")/(1 + 2"v"^2) - "v"`
`x "dv"/("d"x) = (- 2"v")/(1 + 2"v"^2) - "v"`
`x "dv"/("d"x) = (-2"v" - "v"(1 + 2"v"^2))/(1 + 2"v"^2)`
`x "dv"/("d"x) = (-2"v" - "v" - 2"v"^3)/(1 + 2"v"^2)`
`x "dv"/("d"x) = (-3"v" - "v"^3)/(1 + 2"v"^2)`
`int((1 + 2"v"^2)/(3"v" + 2"v"^3)) "dv" = - int ("d"x)/x`
Multiply and Divide by 3, we get
`1/3 int(3 + 6"v"^2)/(3"v" + "v"^3) "dv" = -int ("d"x)/x`
`1/3 log(3"v" + "v"^3) = - logx + log |"C"_1|`
`1/3 log(3"v" + 2"v"^3) + logx = log |"C"_1|`
log (3v + 2v3) + 3log (x) = 3 log (C1)
log (3v + 2v3) + log (x)3 = log (C1)3
log (3v + 2v3)x3 = log C13
(3v + 2v3)x3 = C13
`(3(y/x) + 2(y/x)^3)x^3` = C13
`((3y)/x + (2y^3)/x^3)x^3` = C13
`((3x^2y + 2y^3)x^3)/x^3` = C13
3x2y + 2y3 = C13
3x2y + 2y3 = C is a required solution.
APPEARS IN
संबंधित प्रश्न
If F is the constant force generated by the motor of an automobile of mass M, its velocity V is given by `"M""dv"/"dt"` = F – kV, where k is a constant. Express V in terms of t given that V = 0 when t = 0
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:
`sin ("d"y)/("d"x)` = a, y(0) = 1
Solve the following differential equation:
(ey + 1)cos x dx + ey sin x dy = 0
Solve the following differential equation:
`("d"y)/("d"x) - xsqrt(25 - x^2)` = 0
Solve the following differential equation:
`(x^3 + y^3)"d"y - x^2 y"d"x` = 0
Solve the following differential equation:
`(y^2 - 2xy) "d"x = (x^2 - 2xy) "d"y`
Choose the correct alternative:
The number of arbitrary constants in the particular solution of a differential equation of third order is
Solve: `("d"y)/("d"x) = "ae"^y`
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`
Solve the following:
`x ("d"y)/("d"x) + 2y = x^4`
Choose the correct alternative:
The integrating factor of the differential equation `("d"y)/("d"x) + "P"x` = Q is
Choose the correct alternative:
The differential equation of y = mx + c is (m and c are arbitrary constants)
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:
The solution of the differential equation `("d"y)/("d"x) = y/x + (f(y/x))/(f"'"(y/x))` is
Solve (x2 + y2) dx + 2xy dy = 0
Solve `("d"y)/("d"x) + y cos x + x = 2 cos x`
Solve `("d"y)/("d"x) = xy + x + y + 1`
