Question

If $v=f(x,y)$ is a homogenous function of degree $n$, then which of the follwoing statements is true?

$I.$ $x\frac{\partial v}{\partial x}+y\frac{\partial v}{\partial y}=nv$

$II.$ $x^2\frac{{\partial }^{2}v}{\partial x^2}+2xy\frac{{\partial }^{2}v}{\partial x\partial y}+y^2\frac{{\partial }^{2}v}{\partial y^2}=n(n-2)v$

$III.x\frac{{\partial }^{2}v}{\partial x^2}+y\frac{{\partial }^{2}v}{\partial x\partial y}=(n-1)\frac{\partial v}{\partial x}$

• A. only $I$ is true
• B. only $II,III$ are true
• C. only $I$ and $III$ are true
• D. $I,II,III$ are true

Answer 1

The correct option is C only $I$ and $III$ are true

Given that $v=f(x,y)$ is homogenous of degree n

let $v=f(x,y)=a_0{x}^n+a_1{x}^{n-1}y.1+....+a_{n-1}x{y}^{n-1}+a_n{y}^n$

$\frac{\partial v}{\partial x}=n{a}_0{x}^{n-1}+(n-1)a_1{x}^{n-2}y+.....+a_{n-1}{y}^{n-1}+0$

$\frac{\partial v}{\partial y}=0+a_1{x}^{n-1}y+.....+a_{n-1}(n-1)x{y}^{n-2}+n{a}_n{y}^{n-1}$

$y\frac{\partial v}{\partial y}=0+a_1{x}^{n-1}y+....+a_{n-1}(n-1)x{y}^{n-1}+n{a}_n{y}^n$

$x\frac{\partial v}{\partial x}+y\frac{\partial v}{\partial y}=n{a}_0{x}^n+n{a}_1{x}^{n-1}y+....+n{a}_{n-1}x{y}^{n-1}+n{a}_n{y}^n$

$x\frac{\partial v}{\partial x}+y\frac{\partial v}{\partial y}=nv\to \left(1\right)$

Partially differentiate (1) w.r.t x and y separately

$x\frac{{\partial }^{2}v}{\partial x^2}+\frac{\partial v}{\partial x}+y\frac{{\partial }^{2}v}{\partial x\partial y}=n\frac{\partial v}{\partial x}\to \left(2\right)$

$x\frac{{\partial }^{2}v}{\partial x\partial y}+\frac{\partial v}{\partial y}+y\frac{{\partial }^{2}v}{\partial y^2}=n\frac{\partial v}{\partial y}\to \left(3\right)$

$\left(2\right)\ast x+\left(3\right)\ast y\Rightarrow x^2\frac{{\partial }^{2}v}{\partial x^2}+2xy\frac{{\partial }^{2}v}{\partial x\partial y}+y2\frac{{\partial }^{2}v}{\partial y^2}=n(n-1)v$

$(\because x\frac{\partial v}{\partial x}+y\frac{\partial v}{\partial x}=nv)$

$\left(2\right)\Rightarrow x\frac{{\partial }^{2}v}{\partial x^2}+y\frac{{\partial }^{2}v}{\partial x\partial y}=(n-1)\frac{\partial v}{\partial x}$

In given options, Statement -I and III are true