TITLE: The equation $y'=1+y^4$ blows up in finite time
QUESTION [6 upvotes]: Prove that the equation $$y'=1+y^4 \:, y(0)=0$$ has a maximal solution with a bounded domain. 
The existence and unicity of such a maximal solution is granted by Cauchy-Lipschitz theorem. 
I tried a proof by contradiction for the boundedness. One promising way would be to use the fundamental theorem of calculus and taking limits at infinity... 
Thanks for your suggestions.

REPLY [6 votes]: Here is an indirect solution. If $y$ does not blow up in finite time, then the maximal solution is defined on all of $[0, \infty)$. We make the following two easy observations:

Sine $y' \geq 1$, we get $y(t) \geq t$.
Since $y' = 1 + y^{4} \geq 2y^{2}$, for any $0 < t_{0} < t_{1}$ we get
$$ \frac{y'}{y^{2}} \geq 2 \quad \Longrightarrow \quad \frac{1}{y(t_{0})} - \frac{1}{y(t_{1})} \geq 2(t_{1} - t_{0}). $$

Combining two observations, for any $0 < t_{0} < t_{1}$ we get
$$ \frac{1}{t_{0}} \geq \frac{1}{y(t_{0})} - \frac{1}{y(t_{1})} \geq 2 (t_{1} - t_{0}). $$
Then we get a contradiction by choosing large $t_{0}$ and $t_{1} = t_{0} + 1$.