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These are replies submitted by vv

@Carl Love The fact that

sys:=[a*x1^3 + b*x1^2 + c*x1 + d, a*x2^3 + b*x2^2 + c*x2 + d, a*x3^3 + b*x3^2 + c*x3 + d, x1*x2*x3 = -d/a, x1 + x2 + x3 = -b/a, x1*x2 + x1*x3 + x2*x3 = c/a]:
solve(sys, {x1, x2, x3}, explicit);

does not work, is a weakness of solve.
It is easy to see (but not for a CAS) that sys is equivalent to sys[4..6], which works!

@Carl Love 

g := (x, T) -> T*x + x^2:  #just an example
dgdx1:=(x,T) -> D[1](g)(x,T);
dgdx2:=  D[1](g);
dgdx1(1, 2);  # 4
dgdx2(1, 2);  # 4

g := (x, T) -> T*x + 7*x^2:
dgdx1(1, 2); # 16
dgdx2(1, 2); # 4

For a fixed g, dgdx2 is of course more efficient, but I don't think this matters here.

Your odetest examples are not correctly treated. You cannot use asympt when the series solution is around a finite point (e.g. 'point'=0). asympt does a series expansion around infinity!
Note also that not all ode solutions have series expansions; e.g. for Ex 6, the series can be obtained only with MultiSeries:-series,
so, for x>0 (series and dsolve cannot do it).

@acer  Of course, but that is what the user has asked.

@Axel Vogt Your last command should be:
MultiSeries:-asympt(%, x, 6);
or simply,
asympt(%, x, 6);

Actualy, odetest(sol, ode)  does essentially the same thing.


@nm All your "positive" examples are regular series or Puiseux series.
The solution of the ode in the question is not of these types, and I suspect that odetest rejects it.
But the asympt approach is acceptable in my opinion and works.

@smithss  This one cannot have a solution: there are more columns than colors!

@Carl Love Actually the max number of permutations used is small:

(9!/(3!*2!*2!)) * 6 = 90720

because once a permution if found for a column, it is kept.
The puzzle can be easily solved by hand.

It would be interesting to find a puzzle which cannot be solved this way!

@C_R It would not be a good idea to simply return infinity, because the arithmetic with oo is complicated.
This is why there is try ... end try.  See also ?events

  catch "numeric exception: div":
end try;


@jud What I get is:

@jud  Yes, it seems that in newer versions, DrawSubgroupLattice outputs numbers (1,2,...) instead of symbols (`1`,`2`,...); unfortunately I have not tested Carl's code before posting.
You will have to try my approach which works.

@jud Because you are using the document mode, create a prompt and paste the code there.

@ecterrab Very nice, congratulations!

I was curious to see why Sum=add is needed (instead of value).
It seems that sum has problems with pochhammer. If we replace pochhammer with P,  (and then back), value works.
Here is an example:

S:=Sum(pochhammer(2*k - n + 1, 2*n - 2*k)*pochhammer(3 - k, k)/((n - k)!), k = 0 .. n):
value(%); # ?
#                               0
seq( value(eval(S,n=N)), N=1..6 );
#                       2, 6, 12, 24, 0, 0

The result of value(S) should be  2*n*(n - 1)/GAMMA(-n + 5)  (instead of 0)
which is actually obtained using  assuming n::posint

In this case, the correct values can be obtained with limit insted of eval.

(The situation is similar to sum(k, k=1..n) where n is also "assumed" integer).

@ecterrab In order to use this workaround I had to replace ex1 with allvalues(ex1) in ex2 due to the Sum(..., alpha=RootOf(...)). 
BTW, in an older Maple version I have on a small laptop, your first workaround worked for the "assuming" example (applied blindly!); here ex1 contains a hypergeom function, and diff(...) is absent in ex1. It seems to be a better answer (at least in this case). Maple is interesting!

Best regards,


The answer of ex1 contains an inert sum and diff(u, y$p), for p in 0..k.
But in this case (when diff is accepted) a much simpler symbolic answer is

ans := pochhammer(-j,j)*Diff((y^2)^j*(x*y^2+2)^(-1-j),y $ k);

This one is not fragile :-)

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