Problem with qualitative analysis

Question

I have some problem with the following exercise:

Consider the following Cauchy problem:

$$\begin{cases} y'=(y^2-4) \arctan (1+y^2) \\ y(0)=6 \end{cases}$$ I have to show that the maximal solution of this problem exist and it is defined on an interval $(- \infty, \beta)$ with $0< \beta < \infty$. I have to estimate $\beta$.

My work: I try with $y'=0$ and i found that $y=2,y=-2$ are solutions. Consider the part of region with $y=6$ the solution is always increasing. But I don't know to show if exists an asymptote or not. I have the following hint:

$$ 1 \le 1 + y^2 < \infty$$

Thanks to all, and sorry for my bad english.

Answer 1

Hint

Let $f,g$ be two functions with $f(0)=g(0)=6.$ Assume that

$$f'(x)=f(x)^2-4$$ and $$g'(x)\ge g(x)^2-4.$$

• Show that $g\ge f.$
• Show that $f$ there exists $c>0$ such that $\lim_{x\to c^-}f(x)=+\infty.$
• What can say about $g$?