This was actually from my recent exam questioning :
Prove $$f(\sqrt3) < 1 + \frac{\pi}{12}$$ for function $f$ such that $f \in C(1, \infty), \:f(1) = 1$ and $$f'(x) = \frac{1}{x^2 + \left\{ f(x) \right\}^2}.$$
The solution is :
Since $f'(x) > 0$, $f$ is an increasing function in $(1, \infty)$ and $f(x) > f(1) = 1$ where $x > 1.$
Therefore, $$f'(x) = \frac{1}{x^2 + \left\{ f(x) \right\}^2} < \frac{1}{1 + x^2}$$
and
$$\int_{1}^{x} f'(t) dt < \int_{1}^{x} \frac{1}{1 + t^2} dt,$$ $$f(x) - f(1) < \tan^{-1} x - \tan^{-1} 1,$$ $$f(x) < \tan^{-1} x - \frac{\pi}{4} + 1.$$
let $x = \sqrt3$, and we get the identity : $$f(\sqrt3) < 1 + \frac{\pi}{12}.$$
Now I am curious if there exists $f$ such that $$f'(x) = \frac{1}{x^2 + \left\{ f(x) \right\}^2}.$$
Not only about the existence, but I would also like to find the specific function if possible. Wolfram Alpha gives me the plots, but gives no hint about the specific function itself.
