<s>[INST] How to solve the equation $a(f(x))^2 - af(kx)f(x) + xf’(x) = 0$ What function(s) f satisfies the equation 
$$a(f(x))^2 - af(kx)f(x) + xf’(x) = 0$$
where $a$ and $k$ are constants, $k>0$ and $x$ is a real number?
I’m aware that that the function $f(x)=c$ satisfies the equation i.e. $f(x)$ is a constant function. But can it be proven that there is a non constant  function f that can satisfy the equation?
 [/INST] If $f(x) = ux+v$
with $u \ne 0$
then
$\begin{array}\\
0
&=a(f(x))^2 - af(kx)f(x) + xf’(x)\\
&=a((ux+v)^2 - (kux+v)(ux+v)) + xu\\
&=a(u^2x^2+2uvx+v^2 -(ku^2x^2+uv(k+1)x+v^2))+ xu\\
&=a(u^2(1-k)x^2+uv(2-(k+1))x+v^2)+ xu\\
&=au^2(1-k)x^2+(auv(1-k)+u)x+av^2\\
&=au^2(1-k)x^2+u(av(1-k)+1)x+av^2\\
\end{array}
$ 
so $v = 0$
and the coefficient of $x$ is $1$
which doesn't work.
If $f(x)
=cx^n + O(x^{n-1})$ 
then
$\begin{array}\\
0
&=a(f(x))^2 - af(kx)f(x) + xf’(x)\\
&=a(cx^n+O(x^{n-1}))^2 - a(cx^n+O(x^{n-1}))(cx^nO(x^{n-1})) + x(nx^{n-1}+O(x^{n-2}))\\
&=a(c^2x^{2n}+O(x^{2n-1})) - a(c(kx)^n+O(x)^{n-1}))(cx^n+O(x^{n-1})) + x(nx^{n-1}+O(x^{n-2}))\\
&=ac^2x^{2n}+O(x^{2n-1}) - ac^2k^nx^{2n}+O(x^{2n-1})\\
&=ac^2(1-k^n)x^{2n}+O(x^{2n-1}) \\
\end{array}
$ 
so $k^n = 1$.
I'll stop here.
</s>