homogeneous functions

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We will, on the next page, look at a particular type of differential equation connected to the so-called homogeneous function of two variables, which are functions that follow this rule:

f(sx,sy) = {s^k}f(x,y)

I.e., when you multiply the variables by a scale factor, then the result is the same as if you instead had multiplied the function by that scale factor to the power of some constant. That constant, k, is the degree of homogeneity of the function.

The function

f(x,y) = {x^2} + 3{y^2}

is homogeneous of degree 2 since

\begin{gathered}   f(sx,sy) = {(sx)^2} + 3{(sy)^2} =  \hfill \\ \quad  = {s^2}({x^2} + 3{y^2}) = {s^2}f(x,y) \hfill \\ \end{gathered}  

In general, polynomials of the form

f(x,y) = \sum\limits_{k = 0}^n {{a_k}{x^k}{y^{n - k}},\quad n \in \mathbb{N}}  

are homogenous, as is quite easily verified, and in particular

f(x,y) = {(ax + by)^n},\quad n \in \mathbb{N}

and

f(x,y) = a{x^p}{y^q}

A rational function of two homogeneous polynomials is also homogeneous. This is also easy to verify, but I leave that to you. More generally, the quotient of two homogeneous functions is also homogeneous.

Finally, we have the functions of the forms

f\left( {\frac{x}{y}} \right)

or

f\left( {\frac{y}{x}} \right)

that are homogeneous of degree 0, since any scale factor will cancel.

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Next page : Homogeneous differential equations – a first typeLast modified: May 23, 2025 @ 12:02