How determine all functions $f: R \mapsto R$ differentiable, are not identically zero and such that $f(x)= x f^{\prime}(\frac{x}{\sqrt{3}})$? How determine all functions $f: R \mapsto R$ differentiable, are not identically zero and such that $f(x)= x f^{\prime}(\frac{x}{\sqrt{3}})$ for $x \in R$?
One class of solutions is $f(x) = a x + b x^3$ for constants $a,b$.  These are all the solutions that are analytic.  There are also non-analytic solutions such as 
$$ f(x) = \cases{ a x + b_1 x^3 &amp; for $x \ge 0$\cr
                      a x + b_2 x^3 &amp; for $x &lt; 0$} $$
Also, if $r = a + b i$ is any of the (I would guess infinitely many) complex zeros of
the function $g(z) = 3^{z/2} - \sqrt{3} z$ with $a = \text{Re}(r) \ge 1$, 
$ |x|^a \cos(b \log |x|)$ and $ |x|^a \sin(b \log |x|)$ are solutions (defined to be $0$ at $x=0$).  Again you can "mix and match" the solutions for positive and negative $x$, and take linear combinations.
One class of solutions is $f(x) = a x + b x^3$ for constants $a,b$.  These are all the solutions that are analytic.  There are also non-analytic solutions such as 
$$ f(x) = \cases{ a x + b_1 x^3 &amp; for $x \ge 0$\cr
                      a x + b_2 x^3 &amp; for $x &lt; 0$} $$
Also, if $r = a + b i$ is any of the (I would guess infinitely many) complex zeros of
the function $g(z) = 3^{z/2} - \sqrt{3} z$ with $a = \text{Re}(r) \ge 1$, 
$ |x|^a \cos(b \log |x|)$ and $ |x|^a \sin(b \log |x|)$ are solutions (defined to be $0$ at $x=0$).  Again you can "mix and match" the solutions for positive and negative $x$, and take linear combinations.