Mathematical and Computational Applications, Vol. 18, No. 3, pp. 383-391, 2013 TAYLOR COLLOCATION METHOD FOR SOLVING A CLASS OF THE FIRST ORDER NONLINEAR DIFFERENTIAL EQUATIONS Dilek Taştekin, Salih Yalçınbaş and Mehmet Sezer Department of Mathematics, Celal Bayar University, 45140, Muradiye, Manisa, Turkey dilektastekin@gmail.com, salih.yalcinbas@cbu.edu.tr, mehmet.sezer@cbu.edu.tr Abstract- In this study, we present a reliable numerical approximation of the some first order nonlinear ordinary differential equations with the mixed condition by the using a new Taylor collocation method. The solution is obtained in the form of a truncated Taylor series with easily determined components. Also, the method can be used to solve Riccati equation. The numerical results show the effectuality of the method for this type of equations. Comparing the methodology with some known techniques shows that the existing approximation is relatively easy and highly accurate. Key Words- Nonlinear ordinary differential equations, Riccati equation, Taylor polynomials, collocation points. 1. INTRODUCTION Nonlinear ordinary differential equations are frequently used to model a wide class of problems in many areas of scientific fields; chemical reactions, spring-mass systems bending of beams, resistor-capacitor-inductance circuits, pendulums, the motion of a rotating mass around another body and so forth   1, 2 . These equations here also demonstrated their usefulness in ecology, economics, biology, astrophysics and engineering. Thus, methods of solution for these equations are of great importance to engineers and scientists   3, 4 . In this paper, for our aim we consider the first order nonlinear ordinary differential equation of the form 2 2 ()() () () () () ()() () ( )( ( )) () Pxyx Qxy x Rxy x Sxyxy x Tx y x gx         , a x b   (1) under the mixed conditions (a) + (b) = y y    (2) and look for the approximate solution in the form () 0 () () ( -), , ! n N n n n n y c yx y xc y a c b n       (3) which is a Taylor polynomial of degree N at x c  , where   0,1, , n y n N   are the coefficients to be determined. Here      , , , , Px Qx Rx Sx Tx and  gx are the functions defined on b x a   ; the real coefficients ,   and  are appropriate constants. Note that, if () () 0 Sx Tx   in Eq. (1), it is a Riccati Equation ] 8 5 [  .