Editing Picard–Lindelöf theorem (section)

== Example of non-uniqueness ==
To understand uniqueness of solutions, contrast the following two examples of first order ordinary differential equations for {{math|''y''(''t'')}}.<ref>{{cite book |first=V. I. |last=Arnold |authorlink=Vladimir Arnold |title=Ordinary Differential Equations |publisher=The MIT Press |year=1978 |isbn=0-262-51018-9 }}</ref> Both differential equations will possess a single stationary point {{math|''y'' {{=}} 0.}} 

First, the homogeneous linear equation {{math|{{sfrac|''dy''|''dt''}} {{=}} ''ay''}} (<math>a<0</math>), a stationary solution is {{math|''y''(''t'') {{=}} 0}}, which is obtained for the initial condition {{math|''y''(0) {{=}} 0}}. Beginning with any other initial condition {{math|''y''(0) {{=}} ''y''<sub>0</sub> ≠ 0}}, the solution <math>y(t) = y_0 e^{at}</math> tends toward the stationary point {{math|''y'' {{=}} 0}}, but it only approaches it in the limit of infinite time, so the uniqueness of solutions over all finite times is guaranteed. 

By contrast for an equation in which the stationary point can be reached after a ''finite'' time, uniqueness of solutions does not hold. Consider the homogeneous nonlinear equation {{math|{{sfrac|''dy''|''dt''}} {{=}} ''ay''<sup>&thinsp;{{sfrac|2|3}}</sup>}}, which has at least these two solutions corresponding to the initial condition {{math|''y''(0) {{=}} 0}}: {{math|''y''(''t'') {{=}} 0}} and 

:<math>y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\ \ \ \ \ 0 & t \ge 0, \end{cases}</math>

so the previous state of the system is not uniquely determined by its state at or after ''t'' = 0. The uniqueness theorem does not apply because the derivative of the function {{math|&thinsp;''f''&thinsp;(''y'') {{=}} ''y''<sup>&thinsp;{{sfrac|2|3}}</sup>}} is not bounded in the neighborhood of {{math|''y'' {{=}} 0}} and therefore it is not Lipschitz continuous, violating the hypothesis of the theorem.