Euler’s Method

Euler’s Method - Chapter 08.02 Eulers...

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Unformatted text preview: Chapter 08.02 Eulers Method for Ordinary Differential Equations After reading this chapter, you should be able to : 1. develop Eulers Method for solving ordinary differential equations, 2. determine how the step size affects the accuracy of a solution, 3. derive Eulers formula from Taylor series, and 4. use Eulers method to find approximate values of integrals. What is Eulers method? Eulers method is a numerical technique to solve ordinary differential equations of the form ( 29 ( 29 , , y y y x f dx dy = = (1) So only first order ordinary differential equations can be solved by using Eulers method. In another chapter we will discuss how Eulers method is used to solve higher order ordinary differential equations or coupled (simultaneous) differential equations. How does one write a first order differential equation in the above form? Example 1 Rewrite ( 29 5 , 3 . 1 2 = = +- y e y dx dy x in ) ( ), , ( y y y x f dx dy = = form. Solution ( 29 5 , 3 . 1 2 = = +- y e y dx dy x ( 29 5 , 2 3 . 1 =- =- y y e dx dy x In this case 08.02.1 08.02.2 Chapter 08.02 ( 29 y e y x f x 2 3 . 1 ,- =- Example 2 Rewrite ( 29 5 ), 3 sin( 2 2 2 = = + y x y x dx dy e y in ) ( ), , ( y y y x f dx dy = = form. Solution ( 29 5 ), 3 sin( 2 2 2 = = + y x y x dx dy e y ( 29 5 , ) 3 sin( 2 2 2 =- = y e y x x dx dy y In this case ( 29 y e y x x y x f 2 2 ) 3 sin( 2 ,- = Derivation of Eulers method At = x , we are given the value of . y y = Let us call = x as x . Now since we know the slope of y with respect to x , that is, ( 29 y x f , , then at x x = , the slope is ( 29 , y x f . Both x and y are known from the initial condition ( 29 y x y = . Figure 1 Graphical interpretation of the first step of Eulers method. y Step size, h x ) , ( y x True value y 1 , Predicted value 1 x Eulers Method 08.02.3 So the slope at x x = as shown in Figure 1 is Slope Run Rise = 1 1 x x y y-- = ( 29 , y x f = From here ( 29 ( 29 1 1 , x x y x f y y- + = Calling 1 x x- the step size h , we get ( 29 h y x f y y 1 , +...
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This note was uploaded on 05/01/2011 for the course CHBE 2120 taught by Professor Gallivan during the Spring '07 term at Georgia Institute of Technology.

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Euler’s Method - Chapter 08.02 Eulers...

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