This is from section "existence and uniqueness theorm" in ordinary
diff. eq class. please be detailed and complete in your answer!
  
The version of the theorem that has an explicit estimate on the size
of the interval (-h,h).

--Estimate the size of the interval of existence of the following solution:
 
y=y(t) to y'=sin(ty) with y(0)=0.

I put a high price on one problem for a fast (but complete ) response.
Thanks for the help.

-sarah

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Clarification of Question by sarahinny-ga on 19 Sep 2005 16:33 PDT

Please answer this within 24 hrs ...  I will provide a tip for the
person who does. thanks again!

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Clarification of Question by sarahinny-ga on 22 Sep 2005 18:52 PDT

Hi mathtalks,

Thanks so much for your helpful comments! I am in fact using ODE
Boyce/Di (8th ed.) Could you walk me through the problem- would I
differentiate w/ repect to both x, y? I think it's based on Theorem
2.4.2=

Let the functions f , and df/dy be continuous in some rectangle
containing the pt (t0,y0). Then, in some interval t0 - h< t < t0 + h
contained in a < t< B, there is a unique solution y = l(t) of the
initial value problem
y' = f(t,y), y(t0) = y0.

Please try to respond with the next few hours, as I need to understand for my quiz.

Hi, sarahinny-ga:

It sounds as if you have a specific textbook you are working from as
you mention "The version of the theorem that has an explicit estimate
on the size of the interval (-h,h)."

One common textbook for undergraduate ordinary differential equations
is by Boyce and dePrima, but I dare say that treatments by different
authors will vary to some degree.

Estimates of the interval of convergence depend, as you probably know,
on some measures of the "smoothness" of the nonlinear function f, as
in y' = f(t,y).  The general discussion/proof is necessarily
complicated by the fact that f is a function of both the independent
argument t and the (nominally) dependent argument y.

In the example you ask about, y' = sin(ty).  Have you tried to compute
the partial derivatives of f(t,y) = sin(ty) with respect to the two
"arguments"?  Are there some specific issues you found in trying to
understand what the quantities mentioned in the general proof relate
to in this example?

regards, mathtalk-ga

please look above