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?
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.