Most texts define a homogeneous function as one where $f(tx,ty)=t^kf(x,y)$ for some constant $k$. But I have always been confused about this definition for two reasons.
- Many texts provide an example such as $\sqrt{x^2+y^2}$ and say that it is homogeneous of degree 1. But $\sqrt{(tx)^2+(ty)^2}=\sqrt{t^2(x^2+y^2)}=|t|\sqrt{x^2+y^2}$, so does this mean a function is still homogeneous if it factors out the absolute value of $t$?
- Every definition I've seen doesn't specify that $k$ has to be a whole number, but I haven't seen any examples where $k=\frac{1}{2}$ for example. Can $k$ be any real number?
I am wondering if the answer to the first question is simply that when solving a homogeneous differential equation, the sign of $t$ is irrelevant because it is taken care of by the constant in the general solution, or something similar. And for the second question, I am wondering if $k$ is implied to be a whole number, or if we don't see examples where $k=\frac{1}{2}$ simply because they are too hard to work with.
Second question: the constant $k \in \mathbf{R}$. It can be a (possibly, negative) real number.
First question: given $k \in \mathbf{R}$, it is (implicitly) assumed that $t > 0$ to make sure that $t^k$ is properly defined.
Here is a useful tutorial on Homogeneous functions.