Question

For a given set of constants α, β, and γ the functions

ƒ (x, y) =

7𝑥

𝛼𝑦

2𝑥𝛽

and ġ (x, y) =

3𝑥

𝛽−𝛾

𝑥𝛽−𝑦2

are homogeneous of degree 2 and 1 respectively. Determine the values for α, β, and γ.

Answer 1

An equation of the form f(x,y)=0 is said to be the homogeneous equation of degree n, where n is a positive integer, and if for some real number k, we have f(kx,ky)=kⁿf(x,y)

$f(x,y)=\frac{7x^{\alpha}y}{2x^{\beta}}$

$f(kx,ky)=k^2f(x,y)$

$\frac{7k^{\alpha}x^{\alpha}ky}{2k^{\beta}x^{\beta}}=\frac{7k^2x^{\alpha}y}{2x^{\beta}}$

$k^{\alpha-\beta+1}=k^2$

$\alpha-\beta=1$

$ġ (x, y) =\frac{3x^{\beta-\gamma}}{x^{\beta}-y^2}$

$\frac{3k^{\beta-\gamma}x^{\beta-\gamma}}{k^{\beta}x^{\beta}-k^2y^2}=\frac{3kx^{\beta-\gamma}}{x^{\beta}-y^2}$

$k^{\beta-\gamma}=k(k^{\beta}-k^2)$

$k^{\beta-\gamma-1}=k^{\beta}-k^2$