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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:

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

is homogenous of degree 2 since

In general, polynomials of the form

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

and

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

or

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

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