You can use Newton's method or Picard's method
constructively in an algebraic domain. See for
example,
http.cs.berkeley.edu/~fateman/papers/newton.ps
which contains macsyma programs.
Commercial Macsyma has a program "Taylorsolve"
that, I think, does this. It may even have a
"Newton polygon" construction for cases that
need it.
I think your "simple procedure" is related to Picard's method,
and what you can try is, instead of integrate(y^2,x),
do taylor (ratdisrep (integrate (taylor ... n)...)) , n+1)
where the n, n+1 are the order of the expansion of the
series.
RJF
Barton Willis wrote:
>
>>I could generate the series (to the solution of
>>y'=x^2+y^2, y(0)=1) by the more simple procedure
>>(obtained by simply integrating through the
>>DE):
>>
>>y:1
>>for i:1 thru 10 y:1+x^3/3+integrate(y^2,x);
>>expand(y);
>>
>>*if* there was a way of automatically getting all
>>powers of x greater than 10 to be set to zero, or else the iteration
>>gets swamped by unwieldy high powers of x. Can I use Maxima's
>>rules and patterns to do this?
>>
>>Thanks,
>>Alasdair
>
>
> Try this:
>
> (%i1) tellrat(x^11)$
> (%i2) algebraic : true$
> (%i3) y : 1;
> (%o3) 1
> (%i4) for i : 1 thru 12 do y : rat(1+x^3/3+integrate(y^2,x));
> (%o4) DONE
> (%i5) y;
> (%o5)
> (17649*x^10+17120*x^9+16605*x^8+16160*x^7+15540*x^6+15120*x^5+14700*x^4+16800*x^3+12600*x^2+12600*x+12600)/12600
>
>
> Barton
>
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