मराठी

F(x, y) = x2+y2x-y is a homogeneous function of degree 1.

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प्रश्न

F(x, y) = `(x^2 + y^2)/(x - y)` is a homogeneous function of degree 1.

पर्याय

  • True

  • False

MCQ
चूक किंवा बरोबर
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उत्तर

This statement is True.

Explanation:

Because f(λx, λy) = λ1f(x, y).

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पाठ 9: Differential Equations - Solved Examples [पृष्ठ १९१]

APPEARS IN

एनसीईआरटी एक्झांप्लर Mathematics Exemplar [English] Class 12
पाठ 9 Differential Equations
Solved Examples | Q 23. (v) | पृष्ठ १९१

संबंधित प्रश्‍न

Solve the differential equation (x2 + y2)dx- 2xydy = 0


Solve the differential equation :

`y+x dy/dx=x−y dy/dx`


Show that the given differential equation is homogeneous and solve them.

(x2 + xy) dy = (x2 + y2) dx


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  dy - y  dx =  sqrt(x^2 + y^2)   dx`


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`


For the differential equation find a particular solution satisfying the given condition:

(x + y) dy + (x – y) dx = 0; y = 1 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


For the differential equation find a particular solution satisfying the given condition:

`2xy + y^2 - 2x^2  dy/dx = 0; y = 2`   when x  = 1


Which of the following is a homogeneous differential equation?


\[\frac{y}{x}\cos\left( \frac{y}{x} \right) dx - \left\{ \frac{x}{y}\sin\left( \frac{y}{x} \right) + \cos\left( \frac{y}{x} \right) \right\} dy = 0\]

\[\left( x^2 + y^2 \right)\frac{dy}{dx} = 8 x^2 - 3xy + 2 y^2\]

(x2 − 2xy) dy + (x2 − 3xy + 2y2) dx = 0


(x2 + 3xy + y2) dx − x2 dy = 0


\[\left( x - y \right)\frac{dy}{dx} = x + 2y\]

Solve the following initial value problem:
\[x\frac{dy}{dx} - y + x \sin\left( \frac{y}{x} \right) = 0, y\left( 2 \right) = x\]


Find the particular solution of the differential equation \[\left( x - y \right)\frac{dy}{dx} = x + 2y\], given that when x = 1, y = 0.


Which of the following is a homogeneous differential equation?


Solve the differential equation:  ` (dy)/(dx) = (x + y )/ (x - y )`


Solve the following differential equation:

`"y"^2 - "x"^2 "dy"/"dx" = "xy""dy"/"dx"`


Solve the following differential equation:

(x2 + 3xy + y2)dx - x2 dy = 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


Which of the following is not a homogeneous function of x and y.


F(x, y) = `(ycos(y/x) + x)/(xcos(y/x))` is not a homogeneous function.


Solve : `x^2 "dy"/"dx"` = x2 + xy + y2.


The solution of the differential equation `(1 + e^(x/y)) dx + e^(x/y) (1 + x/y) dy` = 0 is


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 ______.


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:

  1. Show that (x2 – y2) dx + 2xy dy = 0 is a differential equation of the type `dy/dx = g(y/x)`. (2)
  2. Solve the above equation to find its general solution. (2)

The solution of the equation `dy/dx = (3x − 4y − 2)/(3x − 4y − 3)` is ______.


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