On the Tzitzeica Curve Equation Lewis R. Williams Fayetteville State University Faculty Mentor: Nicoleta Bȋlă Fayetteville State University Abstract The Tzitzeica curve equation is a nonlinear ordinary differential equation whose solutions are called Tzitzeica curves. The aim of this paper is to present the Tzitzeica curve equation along with new, particular families of Tzitzeica curves. 105 Explorations |Science, Mathematics, and Technology 106 Lewis R. Williams 107 Explorations |Science, Mathematics, and Technology 108 Lewis R. Williams 109 Explorations |Science, Mathematics, and Technology 110 Lewis R. Williams 111 Explorations |Science, Mathematics, and Technology 112 Lewis R. Williams Figure 1 Figure 2 Figure 3 113 Explorations |Science, Mathematics, and Technology Figure 4 Figure 5 References [1] A. F. Agnew, A. Bobe, W. G. Boskoff and B. D. Suceava, Tzitzeica curves and surfaces, The Mathematica Journal, 12(2010), 1–18. [2] G. Bluman, J. Cole, Similarity Methods for Differential Equations, Appl. Math. Sci., Vol. 13, Springer-Verlag New York, Heidelberg, Berlin, 1974. [3] G. Bluman, S. Kumei, Symmetries and Differential Equations, Appl. Math. Sci, Vol. 81, Springer-Verlag New York, Heidelberg, Berlin, 1989. [4] M. Crâșmăreanu, Cylindrical Tzitzeica curves implies forced harmonic oscillators, Balkan J. Geom. Appl., 7(2002), No. 1, 37–42. 114 Lewis R. Williams [5] P. E. Hydon, Symmetry Methods for Differential Equations, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2000. [6] N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations: Volume 1, Symmetries, Exact Solutions, and Conservation Laws, CRC Press, Boca Raton, 1993. [7] P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, Vol. 107, Second Edition, Springer-Verlag, New York, 2000. [8] P. J. Olver, Classical Invariant Theory, London Mathematical Society, Student Texts, 44, Cambridge University Press, 1999. [9] A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series, Springer-Verlag London Limited, 2012. [10] C. Rogers, W. K. Schief, Bäcklund and Darboux Transformations, Geometry and Modern Applications in Soliton Theory, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002. [11] V. Rovenski, Geometry of Curves and Surfaces with Maple, Birkhäuser Boston, 2000. [12] J. Stewart, Multivarible Calculus: Concepts & Contexts, Brooks Cole, Cengage Learning, 2010. [13] G. Tzitzeica, Sur Certaines Courbes Gauches, Ann. de l’Ec. Normale Sup., 28 (1911), 9–32. 115.