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On the solutions of the mathieu’s equation

This paper presents a new approach in which approximated analytical solutions of the Mathieu’s equation are constructed in the finite form. Depending on parameters of Mathieu’s equations general solutions may obtain following behaviors: either bounded almost periodic, or infinitely increased combining with infinitely decreased and or infinitely increased combining with periodic. | Vietnam Journal of Mechanics, VAST, Vol. 30, No. 4 (2008), pp. 219 – 231 Special Issue of the 30th Anniversary ON THE SOLUTIONS OF THE MATHIEU’S EQUATION Dao Huy Bich1 , Nguyen Dang Bich2 , Nguyen Anh Tuan2 1 Vietnam National University, Hanoi, Vietnam 2 Institute for Building Science and Technology, Hanoi, Vietnam Abstract. As shown in [1] solutions of the Mathieu’s equation were classified on three fundamental kinds depending mainly on its parameters. These solutions were constructed in the form of infinite series. This paper presents a new approach in which approximated analytical solutions of the Mathieu’s equation are constructed in the finite form. Depending on parameters of Mathieu’s equations general solutions may obtain following behaviors: either bounded almost periodic, or infinitely increased combining with infinitely decreased and or infinitely increased combining with periodic. 1. INTRODUCTION Consider a Mathieu’s equation x ¨ + ω 2 (k + p cos ωt)x = 0 (1) solutions of which have important role in investigation of stability problems and in searching solutions of other non-linear differential equations. Consequently, since 1947 McLachlan N.W.[1] presented methods for constructing solutions of the equation (1) and showed the stability domain of solutions in plane (k, p) [see 1, pp. 40-41]. These results later were referenced in works of Kauderer 1961 [see 2, pp. 572-573] and Nguyen Van Dao et al. 2005 [see 3, p.123]. Solution kinds of the Mathieu’s equation and their stability domain are fundamental scientific results taking attention of many scientists, but to now there is a little similar research. The stability domain of solution in plane (k, p) was defined by curves of the form k = m2 + α1 p + α2 p2 + α3 p3 + ., m = 0, 1, 2, 3, . (2) where α1 , α2 , α3 . are coefficients to be determined in the solving process. The relations (2) are so-called characteristic relations. For example, the characteristic relation corresponds m=1 [1] 1 1 1 .