Advertisements
Advertisements
Question
F(x, y) = `(sqrt(x^2 + y^2) + y)/x` is a homogeneous function of degree ______.
Advertisements
Solution
F(x, y) = `(sqrt(x^2 + y^2) + y)/x` is a homogeneous function of degree Zero.
APPEARS IN
RELATED QUESTIONS
Solve the differential equation :
`y+x dy/dx=x−y dy/dx`
Show that the differential equation `2xydy/dx=x^2+3y^2` is homogeneous and solve it.
Show that the given differential equation is homogeneous and solve them.
`x^2 dy/dx = x^2 - 2y^2 + xy`
Show that the given differential equation is homogeneous and solve them.
`y dx + x log(y/x)dy - 2x 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
For the differential equation find a particular solution satisfying the given condition:
`[xsin^2(y/x - y)] dx + x dy = 0; y = pi/4 "when" x = 1`
For the differential equation find a particular solution satisfying the given condition:
`dy/dx - y/x + cosec (y/x) = 0; y = 0` when x = 1
Find the particular solution of the differential equation `(x - y) dy/dx = (x + 2y)` given that y = 0 when x = 1.
(x2 − 2xy) dy + (x2 − 3xy + 2y2) dx = 0
(x2 + 3xy + y2) dx − x2 dy = 0
(2x2 y + y3) dx + (xy2 − 3x3) dy = 0
Solve the following initial value problem:
\[x e^{y/x} - y + x\frac{dy}{dx} = 0, y\left( e \right) = 0\]
Solve the following initial value problem:
\[\frac{dy}{dx} = \frac{y\left( x + 2y \right)}{x\left( 2x + y \right)}, y\left( 1 \right) = 2\]
Solve the following initial value problem:
x (x2 + 3y2) dx + y (y2 + 3x2) dy = 0, y (1) = 1
Solve the following initial value problem:
\[\left\{ x \sin^2 \left( \frac{y}{x} \right) - y \right\}dx + x dy = 0, y\left( 1 \right) = \frac{\pi}{4}\]
Find the particular solution of the differential equation \[\left( x - y \right)\frac{dy}{dx} = x + 2y\], given that when x = 1, y = 0.
Show that the family of curves for which \[\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}\], is given by \[x^2 - y^2 = Cx\]
A homogeneous differential equation of the form \[\frac{dx}{dy} = h\left( \frac{x}{y} \right)\] can be solved by making the substitution
Solve the following differential equation:
`"x" sin ("y"/"x") "dy" = ["y" sin ("y"/"x") - "x"] "dx"`
Solve the following differential equation:
y2 dx + (xy + x2)dy = 0
Solve the following differential equation:
(9x + 5y) dy + (15x + 11y)dx = 0
Solve the following differential equation:
(x2 – y2)dx + 2xy dy = 0
State whether the following statement is True or False:
A homogeneous differential equation is solved by substituting y = vx and integrating it
State the type of the differential equation for the equation. xdy – ydx = `sqrt(x^2 + y^2) "d"x` and solve it
A homogeneous differential equation of the `(dx)/(dy) = h(x/y)` can be solved by making the substitution.
If a curve y = f(x), passing through the point (1, 2), is the solution of the differential equation, 2x2dy = (2xy + y2)dx, then `f(1/2)` is equal to ______.
The differential equation y' = `y/(x + sqrt(xy))` has general solution given by:
(where C is a constant of integration)
Find the general solution of the differential equation:
(xy – x2) dy = y2 dx
Read the following passage:
|
An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form `dy/dx` = F(x, y) is said to be homogeneous if F(x, y) is a homogeneous function of degree zero, whereas a function F(x, y) is a homogeneous function of degree n if F(λx, λy) = λn F(x, y). To solve a homogeneous differential equation of the type `dy/dx` = F(x, y) = `g(y/x)`, we make the substitution y = vx and then separate the variables. |
Based on the above, answer the following questions:
- Show that (x2 – y2) dx + 2xy dy = 0 is a differential equation of the type `dy/dx = g(y/x)`. (2)
- Solve the above equation to find its general solution. (2)
