Homogenous functions

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We will, on the next page, look at a particular type of differential equation connected to the so-called homogenous function of two variables that 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 would have 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 homogenous 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 homogenous polynomials is also homogenous. This is also easy to verify, but I leave that to you. More generally, the quotient of two homogenous functions is also homogenous.

Finally, we have that functions of the forms

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

or

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

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

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Next page : Homogenous differential equations – a first typeLast modified: Dec 28, 2023 @ 16:42