Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

? - infinitely differentiable solutions to initial value problems

  1. Apr 23, 2004 #1


    User Avatar


    I am interested to know whether a theory exists that allows to answer the following sort of question.

    Does a solution of initial value problem of second order differential equation is infinitely differentiable on the set of positive real numbers?

    For example,

    1) the solution of {y''=y^2, y(0)=1, y'(0)=0} is a function y(t) that goes to infinity as t approaches 2.9744 ... and thus is not infinitely differentiable.

    2) {y''=-1/y, y(0)=1,y'(0)=0} is not infinitely differentiable as well, since first order derivative goes to infinity as t approaches 1.25...

    3) on the contrary, for {y''=1/y, y(0)=1,y'(0)=0}, y(t) is infinitely differentiable on the set of reals.

    So, is there any theory which helps to answer such sort of questions without explicitely solving an equation or system of equations?
  2. jcsd
  3. Apr 23, 2004 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    1) the solution you suggest is not defined on the real numbers

    2) erm, the equation x^{1/2} has derivative tending to infinity at the origin, that doesn't stop it being defined there

    All you can say is that any solution must be in C^2, and that is sufficient for most purposes I can think of.
  4. Apr 23, 2004 #3


    User Avatar

    Well, I am interested which differential equations define real analytic functions on at least real positive numbers, and hence at first need to know which of them define infinitely differentiable functions (C^infinity).
  5. Apr 23, 2004 #4

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    lipschitz and peano are the names for the uniquness and existence of solutions (in that order) of a differential equation, but i don't recall the exact wording of the result (ie if it is a smooth solution or just a solution).

    there are various results about this kind of thing but i can't think of where to look for a unifying one, sorry.
  6. Apr 23, 2004 #5
    what your problem reminds me of

    Hi, QQ;

    1) Your problem reminds me of a question which puzzles me:
    Imagine that we have a graph for a solution of some DE. I mean, all comparatively precise data for x and y(x). The question is: Does any procedure exist that can give us the quantitive measure of this solution being an analytic function (generally, a superposition of any given set of functions), e.g.: P=74%? If this probability is going up when we increase the quality of input data x and y(x), the solution could be analytic and there is an additional chance that we'll be able to find it.

    2) Have you heard about Painleve property (http://mathworld.wolfram.com/PainleveProperty.html)? I think it could be connected to your problem.

    Best of luck,
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook