@dharr Thanks again.

@dharr Thank you; this works well.

@dharr The leading monomial of 2x^2 is x^2, which means that these two polynomials share the same head term and are grouped together. How does the "sort" command function in this partitioning? Additionally, the "<" relation can represent any arbitrary monomial ordering, such as tdeg or plex.

@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