Euler&acirc;€™s Method

# Euler&acirc;€™s Method - EulerMethod Major:...

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05/01/11 http://numericalmethods.eng.usf.edu 1 Euler Method Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM  Undergraduates

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Euler Method      http://numericalmethods.eng.usf.edu
http://numericalmethods.eng.usf.edu 3 Euler’s Method Φ Step size, h x y x 0 ,y 0 True value y 1 , Predicted value ( 29 ( 29 0 0 , , y y y x f dx dy = = Slope Run Rise = 0 1 0 1 x x y y - - = ( 29 0 0 , y x f = ( 29 ( 29 0 1 0 0 0 1 , x x y x f y y - + = ( 29 h y x f y 0 0 0 , + = Figure 1   Graphical interpretation of the first step of Euler’s method

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http://numericalmethods.eng.usf.edu 4 Euler’s Method Φ Step size h True Value      y i+1 ,  Predicted value y i x y x i x i+1 Figure 2.  General graphical interpretation of Euler’s method  ( 29 h y x f y y i i i i , 1 + = + i i x x h - = + 1
http://numericalmethods.eng.usf.edu 5 How to write Ordinary Differential  Equation Example ( 29 5 0 , 3 . 1 2 = = + - y e y dx dy x is rewritten as ( 29 5 0 , 2 3 . 1 = - = - y y e dx dy x In this case ( 29 y e y x f x 2 3 . 1 , - = - How does one write a first order differential equation in the form of ( 29 y x f dx dy , =

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