The subject of this manuscript is 'Embedded explicit Runge-Kutta type methods for directly solving special third order differential equations y ''' = f (x, y)'. The item contribution to knowledge is actual. The manuscript is important to the journal’s readership and contains significant and novel aim. In addition, the objectives of the study are within the scope of the journal. However, the present form of the report needs revision. The author should consider the following points:

Comments on abstract, title, references

Abstract: the abstract is well structured but can be improved if information about the technique adopted in deriving the proposed methods is included.

Title: the title capture the objectives of the study.

Referencing: more references could be included especially Lambert J. D. 1976 and some others that are relevant to this work.

Comments on introduction/background

Introduction: the literature review can be improved by considering the inclusion of more recent references. Background: the introduction should contain the logical progression of the background upon which this research is built. Methods such as block method which shares similar attributes with RK-methods should also form part of the introduction

Comments on methodology

The methodology is adequate but more details can still be included to aid better understanding of the paper. For instance, the procedure for obtaining equations (18) and (24) could be included. The author should increase the impact of this work by considering the stability analysis of the proposed methods.

Comments on data and results

Results: numerical results: all the examples considered are academic examples, effort should be made to solve applied examples such as Genesio equation and Blasius equation to further highlight the novelty of the work.

Comments on discussion and conclusions

Discussion: well discussed. Conclusion: the conclusion is ok as its answers most of the research questions.