Consider homogeneous differential equation of the following form.
$$ f'(x) = F(f(x)/x) $$ $$ F(z) \neq z $$
Approach that has been suggested to me is to introduce new local coordinates $r = y/x$ and $s = \ln|x|$. Then, differential equation is supposed to be written as follows.
$$ s'(r) = \frac{1}{F(r) - r} $$
I am trying to attempt such idea as rigorous as possible, but I am unable to do it. My intuition tells me that key issues is to do all operations locally, as some things are not well defined globally, but I am not sure if that is the case, and even if yes, how to do it correctly.
I first start by defining what it means to introduce local coordinates. I defined it as given $(x,y) \in \mathbb{R}^2$ there is a rule that tells you exactly which other point in the plane corresponds to it by $(r(x,y), s(x,y)) \in \mathbb{R}^2$.
Question 1: Is this what is meant by local coordinates? Do we (in general) have to assume some properties about such local coordinates? I would think we need smoothness and invertibility, but as can be seen from the example, $s = \ln|x|$ is not invertible (though, it is smooth). On the other hand, it is locally invertible (around from $x = 0 $).
Then, I say, if there is a function $f(x)$ defined for all $x \in \mathbb{R}$ then its graph is represented as set of points $(x, f(x)) \in \mathbb{R}^2$ and it is known that $f'(x) = F(f(x)/x)$. If I go to the other coordinates, it would mean that the graph is given by the following points.
$$ (r(x,f(x)), s(x,f(x))) = (f(x)/x, \ln|x|) $$
Question 2: How to define $s(r)$? I would think that intuitively $s(r) = s(x(r)) = \ln|x(r)|$ but I do not think that it is well-defined. Is it well defined locally? How to approach this? For example, what if $f(x) = x$?
Question 3: Even if $s(r)$ is well defined locally, would this be the correct strategy to obtain differential equation for $s(r)$?
$$ d_r s(x(r)) = d_x s(x(r)) d_r x(r) = \frac{1}{x(r)} d_r x(r) = \frac{1}{x(r)} \frac{1}{d_x r(x(r))}$$ $$ = \frac{1}{f'(x(r)) - f(x(r))/x(r)} = \frac{1}{F(r)-r} $$
Question 4: How do we get solution to the original equation? Intuitively, we would like to transform the plane back to $(x,y)$ coordinates and then read off what $y = f(x)$ (by choosing accordingly $r,s(r)$). When is such choice possible? When is such choice unique (and gives unique result)?