Presentation of Eules s Method II

University of Babylon
College of Engineering
Department of Environmental Engineering
Engineering Analysis I (ENAN 103)







Numerical Solution of Ordinary Differential Equations

Undergraduate Level, 3th Stage



Mr. Waleed Ali Tameemi
College of Engineering/ Babylon University
M.Sc. Civil Engineering/ the University of Kansas/ USA



2016-2017
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Euler’s Method
This method is utilized in solving ordinary deferential equation numerically. This method required an initial condition (y_0,x_0). The first derivative (dy/dx) provides a direct estimation of the slope that leads to a direct estimation of the ODE.
¬The following steps are required in solving ordinary differential equations using Euler’s Method:

The ordinary deferential equation (ODE) dy/dx=f(x,y)
Initial condition (x_0,y_0)
Required y value at a given x value ?(x_n,y_n)
Step size ?_x=x_(i+1)-x_i
Number of steps n=(x_n-x_0)/?_x
f(x_(i+1),y_(i+1) )=y_(i+1)=y_i+f(x_i,y_i )×?_x
Those steps may be summarized in the following table
i x_i y_i y_(i+1)
0 x_0? Initial Condition y_0? Initial Condition y_1=y_0+f(x_0,y_0 )×?_x
1 x_1=x_0+?_x y_1 y_2=y_1+f(x_1,y_1 )×?_x
2 x_2=x_1+?_x y_2 y_3=y_2+f(x_2,y_2 )×?_x
… … … …
Ex1: Find the value of the given ODE at f(0.2)
x+y-dy/dx=0
Initial condition: y(0)=1
Step size ?_x=0.1