Suppose I am given an ode ${dy\over dx}={1\over x^2}f(xy)$ where $f$ is some arbitrary function.
How then does doing the following help solve the equation? :
First I have a vector field $V=x\partial_x-y\partial_y$
I see that this corresponds to the transformation $(x,y)\to (x\exp(p), y\exp(p))$ where $p$ is a parameter.
Then I sought invariant coordinates $q_1,q_2$ of the vector field such that $V(q_1)=1, V(q_2)=0$
In this case I could take $q_1=xy$ and $q_2=\log|x|$. Though there might be other choices that are more suitable?
Please help!