Corliss mentioned to me (in email) a couple days ago that he & Chang, had been asked about the example you gave, y' = y, one of the 1st times they presented their program. They had to admit they could not handle it at that time. So it occurs to me now that perhaps you thought my program could not handle it. That had not occurred to me before. See "diff0.ode" and "diff.ode" in the table:
http://sode.sourceforge.net/omniresults/special_cases_007/table.html
diff0.ode is y'=y-1
diff.ode is y''=y'
Anyway I suspect your method is likely to have trouble with systems of non-linear equations. I haven't tested these yet - you walk before running, but I have plans to.
Dennis J. Darland
dennis.darland at yahoo.com
http://dennisdarland.com/http://dennisdarland.com/dennisdaze/http://dennisdarland.com/philosophy/http://sode.sourceforge.net/
"According to the World Health Organization, the warming of the planet caused an additional 140,000 deaths in 2004, as compared with the number of deaths there would have been had average global temperatures remained as they were during the period 1961 to 1990. This means that climate change is already causing, every week, as many deaths as occurred in the terrorist attacks on September 11, 2001"
-- Peter Singer _Practical Ethics, Third Edition_, p. 216.
--- On Tue, 6/19/12, Richard Fateman <fateman at eecs.berkeley.edu> wrote:
> From: Richard Fateman <fateman at eecs.berkeley.edu>
> Subject: Re: [Maxima] maxima is 100 times slower than maple
> To: "Dennis Darland" <dennis.darland at yahoo.com>, "maxima List" <maxima at math.utexas.edu>
> Date: Tuesday, June 19, 2012, 2:51 PM
> On 6/19/2012 12:25 PM, Dennis Darland
> wrote:
>
> ...
> > C, however, it is not as simple as producing a Taylor
> series for a given function. If you had the function, you
> would already have the solution! I.e. you have equations
> each with a derivative in the LHS (left hand side) and a
> functions on the RHS's also possibly involving derivatives.
> How, from this, cam Maxima determine the Taylor series?
> Here's how.
> Let us say that you have a function f(x)? that solves
> the differential equation f'(x)=f(x)? and also f(0)=1.
> Type into Maxima:
>
> gradef(f(x),f(x));
> tellsimp(f(0),1);
> taylor(f(x),x,0,10);
>
> .... the result is the solution to the DE, namely the
> taylor series for exp(x).
>
>
>
> higher order DEs can be accomodated by making up names for
> first, second, etc derivatives,
> and defining their properties.
>
> This neglects many issues that eventually come up like
> convergence, singularities, etc.
>
> It should not be expensive to evaluate a taylor series with
> numerical coefficients, even in Lisp.
>
> RJF
>