Question

If u = log v and v(x, y) is a homogenous function of degree n, then xux + yuy =
A) n(x+y) B) n C) n 2 D) xy​

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{u=log\,v\;\;and\;v\;is\;a\;homogeneous\;function\;of\;degree\;n}$

$\underline{\textbf{To find:}}$

$\mathsf{x\,u_x+y\,u_y}$

$\underline{\textbf{Solution:}}$

$\underline{\textbf{Euler's theorem:}}$

$\textsf{If f is a homogeneous function of degree n, then}$

$\boxed{\mathsf{x\,\dfrac{\partial\,f}{\partial\,x}+y\,\dfrac{\partial\,f}{\partial\,y}=n\,f}}$

$\mathsf{Consider,}$

$\mathsf{u=log\,v}$

$\implies\mathsf{v=e^u}$

$\textsf{By Euler's theore,}$

$\mathsf{x\,\dfrac{\partial\,v}{\partial\,x}+y\,\dfrac{\partial\,v}{\partial\,y}=n\,v}$

$\mathsf{x\,\dfrac{\partial(e^u)}{\partial\,x}+y\,\dfrac{\partial(e^u)}{\partial\,y}=n(e^u)}$

$\mathsf{x\,e^u\,\dfrac{\partial\,u}{\partial\,x}+y\,e^u\,\dfrac{\partial\,u}{\partial\,y}=n(e^u)}$

$\textsf{Divide bothsides by}\;\mathsf{e^u}$

$\mathsf{x\,\dfrac{\partial\,u}{\partial\,x}+y\,\dfrac{\partial\,u}{\partial\,y}=n}$

$\implies\boxed{\mathsf{x\,u_x+y\,u_y=n}}$

$\underline{\textbf{Answer:}}$

$\mathsf{Option\;(c)\;is\;correct}$

$\underline{\textbf{Find more:}}$

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