Suppose for all $x$, $$f’(x)/g’(x) = f(x)/g(x),$$ Then what can we say about the relationship between $f(x)$ and $g(x)$?
I think the only solution is $f(x) = cg(x)$ for some non-zero constant $c$. Is there any other possible relationship?
Suppose for all $x$, $$f’(x)/g’(x) = f(x)/g(x),$$ Then what can we say about the relationship between $f(x)$ and $g(x)$?
I think the only solution is $f(x) = cg(x)$ for some non-zero constant $c$. Is there any other possible relationship?
We can obtain $f'/f=g'/g$ from the problem statement. Integrating both sides gives:
$$\ln(f(x))=\ln(g(x))+C_1,$$
where $C_1$ is our combined constant of integration. Exponentiating both sides gives us:
$$f(x)=cg(x),$$
where $c=e^{C_1}$. Your idea was correct.
EDIT: I saw JohnDoe's comment after posting my answer.
EDIT 2: Of course, this all only works when none of $g, g',$ or $f$ is zero.
Then the numerator of quotient rule for derivative of $f/g$ vanishes and so $f/g$ constant. Need to do more work on it for proof...