Determining a homogenous differential equation. Why is it that $$f(x,y)=\frac {\cos x^2}{x+y}$$ is not considered homogeneous?

That is not a "homogeneous function" for the reason Andre Nicolas gives. Another way of looking at it cannot be put as a function of the single variable y/x.  If you divide both numerator and denominator by x you get $\frac{\frac{1}{x}cos(x^2)}{1+ \frac{y}{x}}$ and then we cannot take that "x" in the numerator into the cosine.  But it is also not a differential equation!  Did you mean $\frac{dy}{dx}= \frac{cos(x^2)}{x+ y}$
