Which of the following functions are homogene | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) function $f(x, y)$ is homogeneous of degree $n$ if $f(\lambda x, \lambda y) = \lambda^n f(x, y)$.$A) f(x, y) = x \sin y + y \sin x$$f(\lambda x, \

Vedclass · Accepted Answer

Both $(B)$ and $(C)$

Vedclass · Answer

(D) function $f(x, y)$ is homogeneous of degree $n$ if $f(\lambda x, \lambda y) = \lambda^n f(x, y)$.$A) f(x, y) = x \sin y + y \sin x$$f(\lambda x, \lambda y) = \lambda x \sin(\lambda y) + \lambda y \sin(\lambda x) = \lambda(x \sin(\lambda y) + y \sin(\lambda x))$.Since this is not equal to $\lambda^n f(x, y)$,it is not homogeneous.$B) f(x, y) = x e^{y/x} + y e^{x/y}$$f(\lambda x, \lambda y) = \lambda x e^{\lambda y / \lambda x} + \lambda y e^{\lambda x / \lambda y} = \lambda(x e^{y/x} + y e^{x/y}) = \lambda^1 f(x, y)$.This is a homogeneous function of degree $1$.$C) f(x, y) = x^2 - xy$$f(\lambda x, \lambda y) = (\lambda x)^2 - (\lambda x)(\lambda y) = \lambda^2 x^2 - \lambda^2 xy = \lambda^2(x^2 - xy) = \lambda^2 f(x, y)$.This is a homogeneous function of degree $2$.Therefore,both $(B)$ and $(C)$ are homogeneous functions.

Which of the following functions are homogeneous?

Similar Questions

The particular solution of the differential equation $(x+y) dy + (x-y) dx = 0$ at $x=1, y=1$ is

The slope of the tangent at any point $(x, y)$ on a curve $y = y(x)$ is $\frac{x^2+y^2}{2xy}$,where $x > 0$. If $y(2) = 0$,then a value of $y(8)$ is

The solution of the differential equation $x y^2 d y - (x^3 + y^3) d x = 0$ is

Let $y=y(x)$ be the solution of the differential equation $\left((x+2) e^{\left(\frac{y+1}{x+2}\right)}+(y+1)\right) d x=(x+2) d y$ with the initial condition $y(1)=1$. If the domain of $y=y(x)$ is an open interval $(\alpha, \beta)$,then $|\alpha+\beta|$ is equal to $......$

The general solution of the differential equation $\frac{d y}{d x}=\frac{3 x+y}{x-y}$ is (where $C$ is a constant of integration.)

Vedclass Test Series

Exam Paper Generator

Online Exam Module