? - infinitely differentiable solutions to initial value problems

[QQ]

Hi,

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?

 

[matt grime]

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.

 

[QQ]

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

Ok, here it is defined at least on interval (0, 2.97...),
but that is the question, for which differential equations the solution is defined on all real positives and is infinitely differentiable on all real positives?

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

Yes, function is defined, but not the derivative!

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

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).

 

[matt grime]

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.

 

[Max0526]

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,
Max.