On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application
, Andrew R. Cameron 2,†, Keegan L. A. Kirk 1,†, Nasser Saad 1,*,†, Patrick Strongman 1,† and Nikita Volodin 1,†Round 1
Reviewer 1 Report
I think that the Abstract is very long, it may be appropriate to write a less long and more precise Abstract.
Author Response
- I think that the Abstract is very long, it may be appropriate to write a less long and more precise Abstract.
The title has been change in response to this comment and other reviewers' comments.
Reviewer 2 Report
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Comments for author File: 
Comments.pdf
Author Response
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Author Response File: Author Response.pdf
Reviewer 3 Report
The manuscript is concerned with the polynomial solution of second order linear differential equations with polynomial coefficients.
The introduction, the organization and the bibliography are fine.
However, there are mistakes and improvements are necessary. Here are some of my suggestions.
The title should be improved by including a reference to "second-order" and "polynomial coefficients".
There are too many authors which is not common for this category of manuscripts (in mathematics, the size, the difficulty, etc. ).
The main equation (3) is repeated 7-8 times in the manuscript. Write it less times and make reference to it.
In Section 3 there are many mistakes. All equations should be checked and rewritten. For example, Latin letters a and b are confused with Greek letters α and β, c1 is used instead of γ1, letter z instead of the letter r, the Greek letter ε instead of the Greek letter γ, etc. Explain the notations of the type 2F1, 3F2, G, etc. Reformulate equations (24), (31), (32). etc to be readable.
Omit subsection with reference to Mathematica.
Reformulate equations (64), (65), etc. to be clear.
Improve conclusions.
Author Response
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Author Response File: Author Response.pdf
Reviewer 4 Report
In the present work, the authors established the necessary and sufficient conditions for the existence of polynomial solutions to the linear differential equation.
they show by example that for n ≥ 3, the necessary condition is not enough to
2 ensure the existence of the polynomial solutions. Using Scheffé’s criteria, they show that from this 3 differential equation there are n-generic equations solvable by a two-term recurrence formula. The closed-form solutions of these generic equations are given in terms of the generalized hypergeometric functions.
The results are new, correct and detailed. The methods are well described. The discussion and conclusions are adequately supported by the data. The writing is acceptable.
Taking the above into consideration, I recommend the paper for publication after some minor corrections.
I suggest the following improvements:
If possible, the author is strongly suggested to make some remarks on the asymptotic and oscillatory behavior of the solutions of a class of delay differential equations in the introduction or in the conclusion for the future work.
1) Bazighifan, O. Some New Oscillation Results for Fourth-Order Neutral Differential Equations with a Canonical Operator.Mathematical Problems in Engineering,Volume 2020, Article ID 8825413, 7 pages, https://doi.org/10.1155/2020/8825413
2) R.P. Agarwal, Ch. Zhang, T. Li, Some remarks on oscillation of second order neutral differential equations, Appl. Math. Compt. 274 (2016)178–181.
3. All the references should be in abbreviation according to the “Algorithms”, and they should be arranged in standard form of the current submitted journal.
4. Moreover, the author should further polish the English Language in the whole paper.
Author Response
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Author Response File: Author Response.pdf
