Determining a homogenous differential equation.

Question

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

Answer 1

Because in general we do not have $$\cos\left(\frac{(tx)^2}{tx+ty}\right)=t^k\cos\left(\frac{x^2}{x+y}\right).$$

Edit: The function has been reinterpreted by an editor. The basic reason remains the same, details are slightly different.

Answer 2

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 $\frac{dy}{dx}= \frac{cos(x^2)}{x+ y}$

Answer 3

Yes, that can be written as $cos((1/(y/x))^2)$ so as a function of y/x only. Or, using Andre Nicolas' method, replace y and x by yt and xt t get $cos((tx)^2/(ty)^2)= cos(t^2x^2/t^2y^2)= cos(x^2/y^2)$