Try to solve the equation \[ c_1 \sqrt{f(x)} + c_2f'(x) = c_3 \sqrt{f(x)} f''(x) \] holds for all $x \ge 0$. There might be another condition: $f(0) = 0$.
It is introduced from a high school physics exam problem on $s, v, a$. The answer to the problem makes a hypothesis that the motion is uniformly accelerated motion and checks and says that it is true. It is equivalent to only check when $f(x) = (\alpha x + \beta)^2$ where $\alpha, \beta \ge 0$, then the equation becomes \[ c_1 (\alpha x + \beta) + 2c_2 \alpha (\alpha x + \beta) = 2c_3 \alpha (\alpha x + \beta) \] and find a solution with $\alpha, \beta \ge 0$, saying proved. I don't think it's a rigorous proof.
I wonder whether the equation can be solved rigorously?
Thanks for help.