On Euler’s And Milne’s Linear Multi Step Methods of Solving the Ordinary Differential Equations [2010MSC: 65XX]
Eziokwu, C. Emmanuel *
Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria.
Okereke, N. Roseline
Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
This work discusses the Euler’s and Milne’s linear multistep methods for solving initial value problems of the ordinary differential equations. By this for the Euler’s we seek an approximation 
and for the integral equation 
with a truncation error
. Thereby providing an enhancement accuracy and for the Milne’s method for the integral equation
the iterative formula: 
Converges faster where the local truncation error 
is sufficiently negligible compared to that of Euler. Hence, the milne’ method is a more accurate approximation method than the Euler’s method. Section three points to the fact that linear multi step method so far discussed is convergent provided it is consistent and stable. This result is achievable if the root conditions are satisfied.
Keywords: Continuous functions, ordinary differential equations iterations, integral equations, initial value problems, convergence