Frankly, I am puzzled by what you write. You say that "the differential equation is "homogeneous" if f(tx, ty)= f(x,y) but
(1) you don't write your equations in the form (the one that is given in that form you change to ).
(2) you never replace x and y with tx and ty.
If you are given a definition use that definition!
The first equation you give is . That can be written in " " form as (for some reason you multiply both sides by , changing it from the form you say you want). The "f(x,y)" here is, of course, . Replace x by tx and y by ty: .
Yes, it is also true that a differential equation is "homogeneous" if and only if f(x,y) can be written as a function of the single variable u= x/y. In the case above, . Since there is a single "y" in the numerator and we want that divided by "x", an obvious first step is to divide both numerator and denominator by x:
The others can all be done in much the same way.