Answer to Question #210748 in Differential Equations for Sarita bartwal

Question

The function f: R^2~{(x,y)|y=0 or y= 5x} →R

defined by

f(x,y)= x^2-3xy/(y^2+5yx)

is a homogeneous function

True or false with full explanation

Answer 1

TRUE

Explanation:

f(x, y)= $\dfrac{x^2 - 3xy }{y^2 + 5xy}$

To check whether f(x, y) is homogenous or not we replace x $\rightarrow$ $\lambda$x and y $\rightarrow$ $\lambda$y

and check if f( $\lambda$x, $\lambda$y) = f(x, y).

On replacing x $\rightarrow$ $\lambda$x and y $\rightarrow$ $\lambda$y, we have

f($\lambda$x, $\lambda$y) = $\dfrac{\lambda ^2x^2 - 3\lambda ^2xy }{\lambda ^2y^2 + 5\lambda ^2xy}$

Taking $\lambda ^2$ common from numerator and denominator we have

f($\lambda$x, $\lambda$y) = $\dfrac{x^2 - 3xy}{y^2 + 5xy}$ = f(x, y)

Hence, we see that f( $\lambda$x, $\lambda$y) = f(x, y). So, f(x, y) is homogenous.