Ordinary Differential Equations/Global uniqueness of solution over interval

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If there is local uniqueness to a solution to an IVP (such as implied Picard–Lindelöf theorem), and if we restrict ourselves to solutions over intervals, then there is global uniqueness of solutions.


Theorem. Uniqueness over intervalsEdit

Theorem If solutions over intervals coincide at a single point then they are the same


  1.   and   are solutions to an IVP
  2.   and   are locally unique solutions (by the Picard–Lindelöf theorem for example)
  3. the domains of   and   are both intervals (which contain  , otherwise the initial condition makes no sense)


  1.   and   coincide inside their common domain of definition:  
  2. If  ,  

is also a solution, with domain  . This notation is unambiguous because of the above hypothesis.

Example. y'=y in various interval domainsEdit


Take the IVP  . Therefore  .





Then   and   satisfy all the hypothesis of the theorem

  1. both are solutions to the IVP
  2. both are locally unique because  satisfies the Picard–Lindelöf theorem in all of its domain (   and therefore is also locally Lipschitz)
  3.   and   are both intervals   and   respectively.

Then we observe all of our conclusions:

  1. Inside  ,  
  2. If we fix the intervals   and  , then   and   are the only solutions with exactly those domains

is also a solution to the IVP with domain  .

Remark. Different domains, different functionsEdit

Remark Different domains mean completely different functions.

Remember from set theory that a function simply a set of ordered pairs. For example



are two functions so that   and  , and  ,   and  . Note that they coincide in the intersection of their domains:  

However they are not equal. Remember that two sets are equal iff the have exactly the same elements, which is obviously not the case for   and   since   but  . Therefore   and   are two completely different sets, and therefore two completely different solutions.

The same goes for two functions such as



Many times the domain of a function is implicit, and we forget about it, usually taking the largest possible. But sometimes taking the largest possible domain may not be appropriate. For example when solving differential equations, taking a domain that is too large (and not an interval) may not lead to uniqueness, which is undesirable. In those cases it is necessary to specify very well what domain we are talking about.

Counter-example. Not an interval.Edit

Counter example

Take the IVP  . Look at the infinite family solutions


  for three values of a

which are each determined by any value of a ( =(y(3) ).

Those solutions satisfy all the conditions of the theorem, except that their common domain   is not an interval. Then we observe that all the conclusions fail for  

  1. Inside  ,  
  2. Both have the same domain, but  , but  

All of this happens because the uniqueness of the initial condition cannot propagate from   to the other side of the domain  .