Solutions of a differential and its related integral equation

We are looking for solutions to the following differential equation $$tf'(t)-\mu f(\frac{t}{\mu})+\mu f(0)=0\;\; ;\;\;t\in \mu I,$$ where $$I$$ is an interval containing $$0$$, $$\mu\in (0,1)$$ is a constant, $$\mu I=\{ \mu x: x\in I\}$$, and $$f:I\rightarrow \mathbb{R}$$ is a differentiable function.

Note. The functions $$f(t)=kt$$ satisfy the equation, and the related integral equation is $$\int^x\frac{f(\frac{1}{\mu}t)}{t}dt=\frac{1}{\mu}f(x)+f(0)\log|x|\; ; \; x\neq 0.$$

Now,

(1) Can one obtain its general solution?, if no,

(2) Are there some other infinite classes for its solutions?

(3) What about uniqueness conditions for a special solution?

• This is a functional differential equation, setting $t=e^{-x}$ you can transform it into a delay-differential equation. Usually the solution depends on a history function on an interval, here it looks like $f|_{[\mu,1]}$ gives this initial history, not just an initial condition in one point. Feb 9 '21 at 10:07
Let $$u(x)=f(e^{-x})$$. Then $$u'(x)=-e^{-x}f'(e^{-x})=-tf'(t)=−μf(t/μ)+μf(0)=−μ(u(x-\delta)-A)$$δ where $$δ=-\lnμ$$ and $$A=\lim_{x\to\infty}u(x)$$.
The solution formula is $$u(x)=u(0)−μ\int_0^{x-δ}(u(s)-A)\,ds,~~x>δ.$$ The solution is completely determined by the values of $$u$$ on $$[0,δ]$$, or of $$f$$ on $$[μ,1]$$.
Using an exponential trial one finds a solution of $$u(x)=A+Be^{-x}\implies f(t)=A+Bt$$. This is of course only one special solution class among many more.