Homogenous functions

Up a level : Differential Equations
Previous page : To find particular solutions
Next page : Homogenous differential equations – a first type

We will, on the next page, look at particular type of differential equations connected to the so called homogenous function of two variables that are functions that follows 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 polynomial is also homogenous. This is also easy to verify, but I leave that to you. More generally, the quotient of two homogenous function is also homogenous.

Finally we have that function 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.

Up a level : Differential Equations
Previous page : To find particular solutions
Next page : Homogenous differential equations – a first typeLast modified: Mar 20, 2019 @ 20:30