Advertisements
Advertisements
प्रश्न
Solve the differential equation (x2 + y2)dx- 2xydy = 0
Advertisements
उत्तर
(x2 + y2)dx- 2xydy = 0
(x2 + y2) dx = 2xydy
`dy/dx = (x^2 + y^2)/(2xy)`.........(i)
The equation is a homogeneous equation
Let y= vx,
Differentiat ing w.r.t. x, we get,
`dy/dx=v+x(dv)/dx`
`dy/dx=(x^2+y^2)/(2xy) " from "(i)`
`v+x(dv)/dx=(x^2+(vx)^2)/(2x.(vx))`
`v+x(dv)/dx=(1+v^2)/(2v)`
`x(dv)/dx=(1+v^2)/(2v)-v`
`x(dv)/dx=(1+v^2-2v^2)/(2v)`
`x(dv)/dx=(1-v^2)/(2v)`
`(2v)/(1-v^2)dv=1/xdx`.......(ii)
Which is in variables separatable form
∴ Integrating both sides, we get
`int(2v)/(1-v^2)dv=int1/xdx + c_1`
`therefore -log|1-v^2|=log|x|+logc`
`therefore log|x(1-v^2)|=log|c|`
`therefore x(1-v^2)=c`
Resubstituting `v=y/x` we get
`x(1-y^2/x^2)=c`
`x((x^2-y^2)/x^2)=c `
`therefore x^2 - y^2 = cx`, where c is constant
which is the required general solution
APPEARS IN
संबंधित प्रश्न
Solve the differential equation :
`y+x dy/dx=x−y dy/dx`
Find the particular solution of the differential equation:
2y ex/y dx + (y - 2x ex/y) dy = 0 given that x = 0 when y = 1.
Show that the given differential equation is homogeneous and solve them.
`y' = (x + y)/x`
Show that the given differential equation is homogeneous and solve them.
(x – y) dy – (x + y) dx = 0
Show that the given differential equation is homogeneous and solve them.
`{xcos(y/x) + ysin(y/x)}ydx = {ysin (y/x) - xcos(y/x)}xdy`
Show that the given differential equation is homogeneous and solve them.
`x dy/dx - y + x sin (y/x) = 0`
Show that the given differential equation is homogeneous and solve them.
`(1+e^(x/y))dx + e^(x/y) (1 - x/y)dy = 0`
For the differential equation find a particular solution satisfying the given condition:
x2 dy + (xy + y2) dx = 0; y = 1 when x = 1
Find the particular solution of the differential equation `(x - y) dy/dx = (x + 2y)` given that y = 0 when x = 1.
(2x2 y + y3) dx + (xy2 − 3x3) dy = 0
Solve the following initial value problem:
(x2 + y2) dx = 2xy dy, y (1) = 0
Solve the following initial value problem:
(xy − y2) dx − x2 dy = 0, y(1) = 1
Solve the following initial value problem:
x (x2 + 3y2) dx + y (y2 + 3x2) dy = 0, y (1) = 1
Solve the following initial value problem:
\[\left\{ x \sin^2 \left( \frac{y}{x} \right) - y \right\}dx + x dy = 0, y\left( 1 \right) = \frac{\pi}{4}\]
A homogeneous differential equation of the form \[\frac{dx}{dy} = h\left( \frac{x}{y} \right)\] can be solved by making the substitution
Which of the following is a homogeneous differential equation?
Solve the differential equation: x dy - y dx = `sqrt(x^2 + y^2)dx,` given that y = 0 when x = 1.
Solve the following differential equation:
`(1 + "e"^("x"/"y"))"dx" + "e"^("x"/"y")(1 - "x"/"y")"dy" = 0`
Solve the following differential equation:
`"xy" "dy"/"dx" = "x"^2 + "2y"^2, "y"(1) = 0`
Solve the following differential equation:
(x2 + 3xy + y2)dx - x2 dy = 0
State whether the following statement is True or False:
A homogeneous differential equation is solved by substituting y = vx and integrating it
Find the equation of a curve passing through `(1, pi/4)` if the slope of the tangent to the curve at any point P(x, y) is `y/x - cos^2 y/x`.
State the type of the differential equation for the equation. xdy – ydx = `sqrt(x^2 + y^2) "d"x` and solve it
Which of the following is not a homogeneous function of x and y.
F(x, y) = `(x^2 + y^2)/(x - y)` is a homogeneous function of degree 1.
Solve : `x^2 "dy"/"dx"` = x2 + xy + y2.
A homogeneous differential equation of the `(dx)/(dy) = h(x/y)` can be solved by making the substitution.
Let the solution curve of the differential equation `x (dy)/(dx) - y = sqrt(y^2 + 16x^2)`, y(1) = 3 be y = y(x). Then y(2) is equal to ______.
The solution of the differential equation y2 dx + (x2 − xy + y2)dy = 0 is ______.
