@dharr 

Thanks a lot.

@dharr Thank you so much for your email and efforts. There is not any variable y[i] and all variables are x[i]. Thanks again. 

@sursumCorda 

Thanks for the comprehensive explanations.

@Thomas Richard Thank you so much for your reply.

@sursumCorda 

Thank you so much for your response. For example, I want to know the algorithm behind the "MTM:-rref" Maple command for computing the reduced row echelon form of a matrix. How to use the showstat for this?

@dharr 

Thank you so much. In fact, I want to use expand for sin(x+y) and so on. Expansion for (a+b)x=ax+bx is not different both of them are aA1+bA1. I try to expand the trigonometric function before using Dcoeffs. 

@dharr 

Thanks for your reply. Can we obtain the original ODE by saving all derivative and non-derivative terms in a sorted list with their corresponding Ai and using the subs command?

Also, I think there is a minor problem in the last file you sent. Please see the attached file. Why sin(x) is eliminated?draft3.mw

@dharr 

Thank you so much for your response and efforts. This is fine, and the final question is: How can I convert the obtained polynomial back into the original ODE? 

@dharr 

I need a*A4+b*A5 +c. I deal with any variables but usually x is variable.

@dharr

Thank you again. This is okay, but I believe it is not working properly. Please see the attached file.draft2.mw

@dharr 

Thanks again, but I'm sorry, this also doesn't work in Maple 18!

@dharr 

Thank you so much for your answer. But, when I run your implementation an error appears. Please see the attached file.dcoeffs.mw

@sursumCorda 

Thanks, Yes, this is OK but I need an efficient implementation that receives a list of polynomials and returns as above. Also, it is possible that F contains subset F1,...Fm (Homogeneous polynomials ordered in ascending degrees and not necessarily in consecutive degrees). For example, the input is:
F=[F1={x-y}, F2={-x*y*z+z^3, x*y^2+y*z^2-z^3},F3={-x^2*y*z+z^4, x*y^3+y*z^3-z^4}].  We have to multiply F1 by all monomials in degree two in k[x,y,z] and add them to F2. Now, we must multiply F1 by all monomials in degree 4 multiply F2 by any monomials in degree 2, and add them to F3. 

@Joe Riel Thanks.
OK, I will use the homogeneous option to fix it. But, terms=1 means that a monomial, not -y-4!!

@Joe Riel A file attached namely bug.mw. Also, x^2-y-4 is not binomial while I used the following command in my procedure:

f := A[i]^(i+1)+randpoly([op(`minus`({op(A)}, {A[i]}))], terms = 1, coeffs = rand(-4 .. -1), degree = i)