Volume 4 • Issue 5 • 1000247 J Appl Computat Math ISSN: 2168-9679 JACM, an open access journal Open Access Research Article Applied & Computational Mathematics Arficho, J Appl Computat Math 2015, 4:5 http://dx.doi.org/10.4172/2168-9679.1000247 Keywords: Equivalent; Diferential equations; Derivatives; Integrating factor Introduction In this paper, we introduce undetermined functions method to solve linear irst order ordinary diferential equations. A diferential equation is an equation that relates an unknown function and one or more of its derivatives of with respect to one or more independent variables [1]. If the unknown function depends only on a single independent variable, such a diferential equation is ordinary diferential equation. If the unknown function depends only on many independent variables, such a diferential equation is partial diferential equation. An ordinary diferential is linear if it is linear in the unknown function and its derivatives that involve in it. he order of an ordinary diferential equation is the or-der of the highest derivative that appears in the equation [2]. Moreover, diferential equations are classiied into two main categories. he irst one is ordinary diferential equations and the other is partial diferential equations. A solution of a diferential equation in the unknown function y and the independent variable x on the interval I is a function y(x) that satisies the diferential equation identical for all x in I [2]. A solution of a diferential equation with arbitrary parameters is called a general solution. A solution of a diferential equation that is free of arbitrary parameters is called a particular solution [2]. A solution in which the de-pendent variable is expressed solely in terms of the independent variable and constants is said to be an explicit solution. A relation G(x,y) is said to be an implicit solution of an ordinary diferential equation on an interval I, provided there exists at least one function f that satisies the relation as well as the DE on I [1]. Moreover, solution of diferential equations is classiied as trivial and non-trivial solutions, general and particular solutions and explicit and implicit solutions. Our objective is to introduce linear irst order ordinary diferential equations and their solution method. herefore, irst we deine linear irst order ordinary diferential equations. Finally, we derive solution method to solve linear irst order ordinary diferential equations. Motivation Research questions 1) Does solution method for solving irst order linear ordinary diferential equations in general exist? 2) Can we solve irst order linear ordinary diferential equations without applying integrating factor? here exists method for solving linear irst order ordinary diferential equations by applying integrating factor [3]. One can solve linear irst order ordinary diferential equations without applying integrating factor. In this manuscript, we derive solution method for solving linear irst order ordinary diferential equations without applying integrating factor. Linear First Order Ordinary Diferential Equation and its Solution he linear irst order ordinary diferential equation with unknown dependent variable y and independent variable x is deined by a 0 (x)y + a 1 (x)y (1) = g(x). (4.1) he general solution of the equation in equation 4.1 is given by 1 () () () µ µ = ∫ xgx dx a y x (4.2) where 0 1 () () exp( ( ) ) () µ = ∫ a x x dx a x [3]. Here 0 1 () () exp( ( ) ) () µ = ∫ a x x dx a x is called integrating factor of equation in 4.1. Derivation of Undetermined Functions Method for Solving Linear First Order Ordinary Diferential Equations Let’s consider the equation p(x)y = s(x) (5.1) We diferentiate both sides of equation in 5.1 to get that *Corresponding author: Daniel Aricho, Department of Mathematics, Aksum University, Aksum, Ethiopia, Tel: 251347753645, 251348750240; E-mail: daniel.aricho@yahoo.com Received July 31, 2015; Accepted August 17, 2015; Published August 21, 2015 Citation: Aricho D (2015) Undetermined Functions Method for Solving First Order Differential Equations. J Appl Computat Math 4: 247. doi:10.4172/2168- 9679.1000247 Copyright: © 2015 Aricho D. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Abstract Most authors of differential equations used integrating factor to solve linear irst or-der ordinary differential equations. In this paper, we introduce undetermined functions method to solve linear irst order ordinary differential equations. Moreover, we derive solution method for solving linear irst order ordinary differential equations without applying exactness condition. Undetermined Functions Method for Solving First Order Differential Equations Daniel Aricho* Department of Mathematics, Aksum University, Aksum, Ethiopia