I was trying to solve a question in physics that involved a spring that had uniformly distributed mass. To find out how the speed of a small segment of the spring varied from the distance x from the base of the spring I used the fact that the integral of m*v over all the parts of the spring will be equal to mass * velocity of the center of mass. I ended up with this differential equation. However, I have no clue how to solve it.
So, what I need is solutions to $f(x) + x\,f'(x) = f(2x)$.
Based on intuition, I guessed a linear polynomial and it fits the equation but I was wondering if there exist more functions that fit the given equation.
Suppose $$f(x)=\sum_{n=0}^{\infty}a_nx^n$$ Then $$f(x)+x\,f'(x)=\sum_{n=0}^{\infty}a_nx^n+a_1x+2a_2x^2+3a_3x^3+\cdots$$ and $$f(2x)=\sum_{n=0}^{\infty}2^na_nx^n$$ so we get $$a_n+na_n=2^na_n$$ which implies $a_n=0$ for $n\geq 2$.
Conclusion: among all real-analytic functions (a fancy name for real valued functions that have a representation as a power series), only linear functions satisfy the equation.
Now, since the problem comes from physics, it is actually solved.