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

Previous page : To find particular solutions
Next page : Homogeneous differential equations – a first type
