Question: Let $u:\mathbb{R}\to \mathbb{R}$ be a twice continuously differentiable function such that $u(0)>0$ and $u'(0)>0$. Suppose $u$ satisfies$$u''(x)=\frac{u(x)}{x^2+1}\quad \forall x\in \mathbb{R}.$$Prove that the following statements are true:
(1) The function $uu'$ is monotonically increasing on $[0,\infty )$.
(2) The function $u$ is monotonically increasing on $[0,\infty )$.
This was one of the MCQ type question in my exam which I did by exemplifying $u(x)=1+\tan ^{-1}x$, luckily it made me discard the other options. I am looking for the proofs of above statements. I tried to solve it being a second order ordinary differential equation, couldn't do it with existing methods that I know.
Any help? Thanks.
Best Answer
For the first one: $$(u(x)u(x)')'=u(x)'^2+u(x)u(x)''=u(x)'^2+\frac{u(x)^2}{x^2+1}>0$$ for every $x\in [0,\infty)$, so 1) is true.