### Journal Menu >> Advances in Pure Mathematics, 2011, 1, 3-4 doi:10.4236/apm.2011.11002 Published Online January 2011 (http://www.scirp.org/journal/apm) Copyright © 2011 SciRes. APM New Solutions to Nonlinear Ordinary Differential Equations Moawia Alghalith Economics Department, UWI, Trinidad, Cuba E-mail: malghalith@gmail.com Received January 11, 2011; revised January 19, 2011; accepted January 29, 2011 Abstract In contrast to the Euler method and the subsequent methods, we provide solutions to nonlinear ordinary dif-ferential equations. Consequently, our method does not require convergence. We apply our method to a second-order nonlinear ordinary differential equation ODE. However, the method is applicable to higher or-der ODEs. Keywords: Ordinary Differential Equations ODE; Euler ’s Method1. Introduction There are several methods of solving nonlinear ordinary differential equations, such as the Euler method, Runge- Kutta methods and linear multistep methods. For a de-tailed description of these methods, see, for example, Kaw and Kalu (2009), Cellier and Kofman (2008) and Butcher (2008). However, these methods are approxima-tions of the so lution and t h us the y require the ass umptio n of convergence to the solution. Consequently, numerical methods, based on real data, are needed to obtain a solu-tion. In this paper, in contrast to the previous methods, we present solutions to nonlinear ordinary differential equa-tions without the requirement of convergence and with-out the need to numerical methods. In addition, our me-thod is far s imp ler t han the e xisti ng methods. 2. The Model We attempt to solve the following nonl inear or dinary dif-ferential equation (higher-order equations can also be used) ( )( )( )( ), ,,,ysf sysyssS′′ ′= ∈ Consider the following Taylor expansion of y around α ()( )( )()( )()()21,2ys yysysRsα αααα′ ′′=+−+− + where ( )( )()()()( )()21 2Rsys yysysα αααα′=−+ −′′+− is the remainder. Our intermediate goal is to minimize the remainder (in abs olute value) with respect to ti me s ( )minsRs The first-order condition yields ( )( )( )( )( )0,Rsys yysααα∗∗ ∗′′′ ′′=− −−= and thus ( )( )( )( )( )( )21.2ys yysysααα αα∗ ∗∗′ ′′= +−+− (1) Since ( )( )( )( ),,,yfy yααα α′′ ′= we obtain ( )( )( )( )( )( )( )( )21 ,,.2ys yysfy ysααααα αα∗∗∗′=+−′+− But ( )( )( ),y gyα αα′= (this is a result of integrating (1)) and thus subst itut in g this into (1), we obtain ( )( )( )( )( )( )( )( )( )2,1 ,,. 2ysygysfy ysα ααααα αα∗∗∗=+−′+− The initial values ( )( )( ), ,,,y gyαα αα and ( )( )( ),,fy yαα α′ are known (assumed by the pre-vious literature). Thus, this is a solution to (1). The ex-tension of this method to higher-order differential equa-tions is straightforward. M. ALGHALITH Copyright © 2011 SciRes. APM 4 3. Practical Examples For simplicity, we present two first-order numerical ex-amples. Using the above procedure, the solution for a first-order differential equation takes the form ( )( )( )( )( ),,ysyg ysα ααα∗∗=+− (2) and the minimization necessary condition ( )( )( ), 0.ysgyαα∗′−= Example 1. ( )2ys s′= with initial values 1,α= ( )1yα= and ( )( ), 1.gyαα= It is well established in the literature that the solution of the differential equation depends on the initial values, and that different initial values produce different solu-tions. Therefore, from the necessary condition, *21s= and thu s *1s= −, since *sα≠ by construction. Hence, using (2), we obtain ( )( )1 11 11.ys∗= +⋅−−=− Example 2. ( )1yss s′=−+ with in i tial values 0,α= ( )0yα= and ( )( ), 1.gyαα= Hence, **11ss− +=; therefore , *1s= and thus ( )( )0110 1.ys∗= +⋅−= 4. Referen ces  J. C. Butcher, “Numerical methods for ordinary differen-tial equations,” Wiley, W. Sussex, England, 2008.  F. Cellier and E. Kofman, “Continuous system simula-tion,” Springer Verlag, New York, NY, 2006.  A. Kaw and E. Kalu, “Numerical methods with applica-tions,” www.autarkaw.com, 2009.