Maximal interval of existence for IVP Consider the following IVP:
$$x'(t)=2t\cos t+x^2,\quad x(0)=0.$$
I'm trying to figure out whether the maximal interval of existence for the IVP is finite or not. I have tried to apply the comparison theorem as we did with the IVP $x'(t)=t^2+x^2,x(0)=1$, but I'm having difficulty finding the upper/lower solutions.
Any help would be appreciated. Thanks!
 A: The equation is of Riccati type, thus can be analyzed by transforming it into a linear system. Set $x=\frac{v}{u}$, with $u,v\ne 0$, then
$$
uv'-u'v=qu^2+v^2\iff u(v'-qu)=(u'+v)v
$$
can be simplified by demanding $u'=-v$, which implies $v'=qu$ or
$$
u''+qu=0,~~~u(0)=1,~~u'(0)=0.
$$
Poles of $x$ that limit the domain of the maximal solution are found at the roots of $u$. Roots can be shown to exist by employing the Sturm-Picone comparison theorem.
Due to the nature of $q(t)=2t\cos t$ there are "hyperbolic" segments with $q<0$ and "elliptic" segments with $q>0$. Of the elliptic segments we know from the Sturm-Picone comparison theorem that if $q(t)\ge\omega^2$ on an interval with length greater than $\frac{\pi}{ω}$, then this interval contains at least one root of $u$.
Of the intervals where ${\rm sign}(t)\cos(t)\ge \frac12$, the first two in both directions are too small for this argument, but have indeed roots close-by or inside. The second intervals in both directions, $-3\pi+\frac\pi3[-1,1]=[-\frac{10\pi}3,-\frac{8\pi}3]$ and $2\pi+\frac\pi3[-1,1]=[\frac{5\pi}3,\frac{7\pi}3]$, provide a sufficiently large lower bound for $q$, for instance $q(t)\ge 4$ in both cases, so that the necessary interval length $\frac\pi2$ is below the actual interval length $\frac{2\pi}3$. Thus the maximal domain is bounded in both directions.
Numerically one gets that the maximal domain is slightly larger than $(−2.97334,\,1.854655)$.

