Re: [Axiom-developer] A question about differential equations

[daly]

It is rarely the case that algorithms used for hand solutions match
the algorithms used by Axiom (or by computer algebra systems in general).

The Computer Algebra Test Suite (CATS) at:
http://axiom-developer.org/axiom-website/CATS/index.html
has an ODE section (Kamke) which shows Axiom's ODE abilities. 

These are presented in three different forms, 
 * as text files which can be used as input, 
    http://axiom-developer.org/axiom-website/CATS/kamke0.txt
 * as source files which are latex, 
    http://axiom-developer.org/axiom-website/CATS/kamke0.input.pamphlet
 * and as PDFs which are the output of the latex.
    http://axiom-developer.org/axiom-website/CATS/kamke0.input.pdf

There are occasions when the Axiom output did not simplify to the
result given in the textbook. At those times I used RewriteRule to
apply simplifications. 

Look at, for example:

http://axiom-developer.org/axiom-website/CATS/kamke0.input.pamphlet
for the expression ODE6 (--S 24) which gives an example of using RewriteRule
to manipulate the expression.
   ode6 := D(y(x),x) + y(x)*cos(x) - sin(2*x)/2
   ode6a:=solve(ode6,y,x)
   yx:=ode6a.particular
   ode6expr:=D(yx,x) + yx*cos(x) - sin(2*x)/2
   sin2rule := rule 2*cos(x)*sin(x) == sin(2*x)
   sin2rule ode6expr

http://axiom-developer.org/axiom-website/CATS/kamke1.input.pamphlet
Look at ODE94 (in kamke1) where I use
   ode94 := x*D(y(x),x) +a*y(x) + b*x**n
   ode94a:=solve(ode94,y,x)
   yx:=ode94a.particular
   ode94expr := x*D(yx,x) +a*yx + b*x**n
   exprule := rule x^n == %e^(n*log(x))
   exprule ode94expr

I'm not sure the exact form of the result you're trying to achieve
but it is likely that you can use RewriteRule to get there.

Tim


>In putting Axiom through its paces just recently (yes: Axiom, not a fork!),
>I experimented with the ODE
>
>y''+6y'+5y = 10x^2+4x+4exp(-x)
>
>Now standard techniques (such as I teach my students), produce a solution
>of the form
>
>y = A*exp(-5x)+B*exp(-x)+2*x^2-4*x+4+x*exp(-x).
>
>This is Axiom:
>
>--(View this section in a fixed width font if it isn't shown as such)
>
>(1) -> y:=operator 'y
>(2) -> deq:=D(y(x),x,2)+6*D(y(x),x)+5*y(x)=10*x^2+4*x+4*exp(-x)
>(3) -> sol:=solve(deq,y,x)
>   (3)
>   [
>       particular =
>             - x 6       2               - x 5     - 5x  - x     2  - 5x
>       4x (%e   )  + (10x  - 16x + 16)(%e   )  - %e    %e    - 2x %e
>       -----------------------------------------------------------------
>                                       - x 5
>                                   4(%e   )
>     ,
>              - x   - 5x
>    basis= [%e   ,%e    ]]
>Type: Union(Record(particular: Expression(Integer),basis:
>List(Expression(Integer)))
>
>-- 
>of which the particular solution is a bit of a jumble.   It doesn't seem to
>be particularly simplifiable:
>
>(4) -> simplifyExp(sol.particular)
>(4) ->
>           2              - 5x             - 6x
>        (8x  - 16x + 16)%e     + (4x - 1)%e
>   (4)  ---------------------------------------
>                           - 5x
>                        4%e
>
>-- 
>Is there some way I can coerce Axiom into giving a particular solution to
>such a straightforward ODE in a more simplified form?
>
>