Exact Solutions of Nonlinear Second-Order Autonomous Ordinary Differential Equations: Application to Mechanical Systems

Round 1

Reviewer 1 Report

In this paper, a method has been developed for computing exact solutions of second order nonlinear autonomous undamped ordinary differential equations. Illustrative examples considering different types of non-linearity present in classical physical systems are used to further validate the obtained exact solutions, compared with RK4 and Newmark numerical methods. Good agreement has been shown in the validation. The work done in this paper is impressive, which is meaningful in this field. The paper is recommended to be published in Dynamics after answering following questions and finishing corresponding minor revision in the manuscript.

(1)   In Line 349, it should be “For damped systems” instead of “For undamped systems”.

(2)   In Line 74, there is a block symbol. What’s it for?

(3)   Apart from the autonomous undamped requirement, is there any further requirement for the second order nonlinear differential equation to apply the method developed in this work? This could be emphasized in the abstract and conclusion.

none

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

The research subject in the article is interesting, relevant and has many applications in different fields. The article is written in detail and neatly.

I have the only question about obtained small numerical errors (at t>6) in results obtained by RK4 and Newmark with respect to developed exact solution in Fig.4. What is their relative value. Why there are not such numerical errors in Fig.3.  Since both numerical methods have the same shift error, maybe the exact solution is not quite accurate in this case? It would be nice to provide additional clarification.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf