In the most general context, the Picard-Lindelöf theorem (aka Cauchy-Lipschitz in French) asserts the existence of a maximal solution for $\dot{x}(t) = f(t,x(t))$, i.e. of a solution $x(t)$ defined on a interval $I$ such that there exist no other solution whose restriction to $I$ coincide with $x$. The usual proofs of this (when $f$ is such that there is no local unicity) use Zorn's lemma, or some other weaker form of choice. But is this result actually not provable in ZF?



Differential equations and axiom of choice
At least for scalar equations $\dot x(t)=f(t,x(t))$, that is with a nonlinearity $f\in C^0(\Omega,\mathbb{R})$, defined on an open set $\Omega\subset \mathbb{R}^2$, the Zorn's lemma is not necessary: the order structure of $\mathbb{R}$ allows to select a preferred solution (actually, two)


Any IVP admits an upper and a lower solution, whose domains are maximal intervals. For $(t\_0,x\_0)\in\Omega$, define, for $t\in\mathbb{R}$
$$x ^ \* (t):=\sup\big\{x(t)\, : x\in C^1(\mathrm{co}(t\_0,t),\, \mathbb{R}),\, \mathrm{graph}(x)\subset\Omega, x(t\_0)=x\_0, \dot x(s)=f(s,x(s)) \big\}\, ,$$
(where of course $\sup\emptyset=-\infty$). Define
$\mathrm{dom}(x ^ \*)$ to be the connected component of $t\_0$ in the set $\{ t: x ^ \*(t) \in\mathbb{R} \}$. Then, $x ^ \*$ is a solution of the ODE with IVC $x(t\_0)=x\_0$, maximally defined on the interval $\mathrm{dom}(x ^ \*)$.


