I am taking my first ever differential equations course atm and the way we are taught it is just "try to rewrite the equations where all the variables are in the form $\frac{y}{x}$ and if they are do the substitution $z = \frac{y}{x}$.
When I first learnt about Euler's homogeneous functions theorem earlier this semester, we had a clear approach to check whether a function was homogeneous or not. If you had $f(x,y)$, check if $f(tx, ty) = t^kf(x,y)$ then it's a homogeneous function.
These are also called homogeneous function so I could only think they were AT LEAST somewhat similar? Is there any "tips or tricks" that I wasn't taught about testing whether an ODE was homogeneous or not that you guys can hopefully show me?
Thanks in advance I guess and sorry for the long-ish and seemingly clueless post because I really don't know what I am looking for.
The $f(tx,ty)=t^kf(x,y)$ definition is sufficient.
It has the advantage of finding out what $k$ equals.
Then you have the option of dividing through by either $x^k$ and substituting $z=\frac{y}{x}$ or dividing through by $y^k$ and substituting $z=\frac{x}{y}$.