From: israel@math.ubc.ca (Robert Israel) Subject: Re: Why nobody answer this question? Date: 20 Oct 2000 22:05:17 GMT Newsgroups: sci.math Summary: [missing] In article , Zakir F. Seidov wrote: >Given ODE: >(x^2 y')' = -x^2 y^2, >with initial values: >y(0)=1, and y'(0)=0. >Series expansion (get by Mathematica) gives, >for the convergence radius, > the numerical value 3.9626. >What is the meaning of this value? >Is there any general principle governing >that the particular ODE has the particular convergence radius >of the serial expansion of the solution. >There is no pole of y(x) around 3.9626 (or there's?!)) >so I do not understand why the this value?! The solution in the complex plane has some sort of singularity on the circle of radius 3.9626 centred at the origin, if that's the radius. Actually the series, according to Maple, is 2 4 11 6 8 97 10 y(x) = 1 - 1/6 x + 1/60 x - ---- x + 1/8505 x - -------- x + 7560 10692000 457 12 98239 14 13339 16 --------- x - ------------- x + ------------- x - 673596000 1980372240000 3740703120000 3276433 18 20 ----------------- x + O(x ) 12953119728780000 and assuming this pattern of signs continues indefinitely the closest singularities to the origin will be on the imaginary axis. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2