Int. J. Nonlinear Anal. Appl. 11 (2020) No. 1, 199-206 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2018.14644.1769 A new approach for computing the exact solutions of DAEs in generalized Hessenberg forms M. Golchian a , M. Gachpazan b , S. H. Tabasi a,∗ a School of Mathematics and Computer Sciences, Damghan University, P.O.Box 36715-364, Damghan, Iran. b Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran. (Communicated by Madjid Eshaghi Gordji ) Abstract In this paper, we propose a new method, with different approach and economical computing, that presents explicit formulas for the exact solutions of a large class DAEs in Hessenberg forms. First, we illustrate the method for linear time-varying DAEs in Hessenberg forms, in order to show the different approach and also the advantages of the method in computing, that make it economical. Then, we describe that the method is efficient for larger classes including special case of non-linear DAEs in Hessenberg forms. Some examples are given to illustrate the proposed method. Keywords: Differential Algebraic Equations, The Hessenberg Forms, Linear Time-Varying, Linear Systems of Equations, Backward Substitution. 2010 MSC: Primary 65L80. 1. Introduction Differential Algebraic Equations (DAEs) arise in several areas of sciences and engineering. Specially, in last two decades, lots of activities are done for improving the theory and computations of DAEs. In this paper, we are interested in a large class of DAEs in Hessenberg forms of arbitrary size r, that can be defined, both for linear time-varying and non-linear DAEs. The Hessenberg forms arise in many applications for the higher index DAEs. Many of the mechanics and variational problems, are of size two and three [22], and also, some beam deflection problems are of size four [10]. Moreover, according to precise definition of solvability in [4], which is equivalent to existence ∗ Corresponding author Email addresses: mmgolchian@gmail.com (M. Golchian), gachpazan@um.ac.ir (M. Gachpazan), tabasi@du.ac.ir (S. H. Tabasi) Received: April 2018 Revised: November 2018