= Picard-Lindelöf theorem
{c}
{title2=$y'=f(t,y),\quad y(t_0)=y_0$}
{wiki=Picard–Lindelöf_theorem}
= Picard's theorem for ordinary differential equations
{c}
{synonym}
Let $U\subset\mathbb R\times\mathbb R^d$ be an <open set> and $f:U\to\mathbb R^d$ a <continuous function>. Suppose $f$ is <locally Lipschitz> in the state $y$, with a <Lipschitz constant> uniform on a sufficiently small time-state neighborhood of each point. Then each $(t_0,y_0)\in U$ determines a unique local <solution of a differential equation> to the displayed <initial value problem>, on an <interval> around $t_0$. This is the local theorem in <Ordinary Differential Equations and Dynamical Systems by Gerald Teschl>.
For example, $y'=y$, $y(0)=1$, has the <solution of a differential equation> $y(t)=e^t$: its <derivative> is itself and its starting value is $1$. The <Lipschitz constant> is $L=1$, since $|u-v|=|f(t,u)-f(t,v)|$. Thus uniqueness follows even before computing $y(1)=e\approx2.71828183$. <Picard iteration> below constructs this <exponential function> one <polynomial> at a time.
The <Lipschitz condition> has real content. For the <continuous function> involving a <square root>,
$$
f(y)=2\sqrt{\max(y,0)},\qquad y'=f(y),\qquad y(0)=0,
$$
each waiting time $c\geq0$ gives a different forward <solution of a differential equation>:
$$
y_c(t)=\begin{cases}0,&0\leq t\leq c,\\(t-c)^2,&t>c.\end{cases}
$$
The constant <solution of a differential equation> $y(t)=0$ is another choice. At $t=2$, delays $c=0,1,2$ give $y=4,1,0$ respectively. Before the join, both sides of the <ordinary differential equation> are zero; afterwards, $y_c'=2(t-c)=2\sqrt{y_c}$. Both one-sided <derivatives> at $c$ are zero, so the join is valid. All these trajectories can also be extended by zero to $t<0$. The failed <Lipschitz condition> is visible in the quotient
$$
\frac{|f(\varepsilon)-f(0)|}{|\varepsilon-0|}=\frac{2}{\sqrt\varepsilon}\longrightarrow\infty\qquad(\varepsilon\downarrow0).
$$
For $\varepsilon=10^{-2},10^{-4},10^{-6}$ this quotient is $20,200,2000$. <Continuity> still allows local existence by the <Peano existence theorem>; it cannot choose a waiting time. MIT's https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2009/a8c15198f81e85870da85da9792565b9_MIT18_034s09_lec02.pdf[18.034 Lecture 2, Example 2.3] develops the same waiting-time mechanism for a different <ordinary differential equation>, $y'=3y^{2/3}$.
\Image[/ode-uniqueness-and-blowup.png]
{title=Failure of uniqueness and finite-time blowup are different obstructions}
{description=Left: several delayed <solutions of a differential equation> of $y'=2\sqrt{\max(y,0)}$ share $y(0)=0$. Right: the unique <solution of a differential equation> $y=\tan t$ of $y'=1+y^2$, $y(0)=0$, has <finite-time blowup> at $T=\pi/2$. The right vertical axis ends at $10$; the curve continues upward beyond the plot, and there is no finite endpoint value.}
The useful promise is that an <initial condition> and a sufficiently regular local rule select one trajectory. It justifies comparing models and following trajectories without an arbitrary choice at each instant. A locally <Lipschitz continuous> rule need not keep that trajectory bounded: $y'=1+y^2$ meets the theorem's hypotheses everywhere but has <finite-time blowup>. Longer existence is governed by the <finite-dimensional continuation criterion>.
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