Find all continuous functions $ f : \mathbb{R} \to \mathbb{R} $ such that $(f(x)g(x))' = f'(x)g'(x), f,g \neq const $

My solution: $ f'(x)g(x)+g'(x)f(x) = f'(x)g'(x) \\ g(x)+g'(x)f(x)/f'(x) = g'(x) \\ f(x)/f'(x) = (g'(x)-g(x))/g'(x) $

But how to proceed further to reduce to something concrete, I do not know

  • 2
    $\begingroup$ Does $f$ satisfy this property for any $g$ or do you need to find a relation between $f$ and $g$ for a given $g$ ? $\endgroup$
    – Delta-u
    May 22 '18 at 16:35
  • $\begingroup$ The functions are arbitrary, it is supposed to find some general form of such functions. In addition to the constants, of course $\endgroup$ May 22 '18 at 16:39



$$\frac{1}{[\ln f(x)]'}+\frac{1}{[\ln g(x)]'}=1$$

call $\ln f(x)=p(x)$ and $\ln g(x)=q(x)$



  • 1
    $\begingroup$ I like that better than my answer. $\endgroup$ May 22 '18 at 16:59

You can fix a function $f(x)$, and then you obtain a differential equation for $g(x)$, which has a unique solution up to a constant multiple: $$ f'(x)g(x) + g'(x)f(x) = f'(x)g'(x) $$ $$ g'(x) = \frac{g(x)}{1 - \frac{f(x)}{f'(x)}} $$


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