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Euler’s Method

Euler’s Method - Chapter 08.02 Eulers...

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Chapter 08.02 Euler’s Method for Ordinary Differential Equations After reading this chapter, you should be able to : 1. develop Euler’s Method for solving ordinary differential equations, 2. determine how the step size affects the accuracy of a solution, 3. derive Euler’s formula from Taylor series, and 4. use Euler’s method to find approximate values of integrals. What is Euler’s method? Euler’s method is a numerical technique to solve ordinary differential equations of the form ( 29 ( 29 0 0 , , y y y x f dx dy = = (1) So only first order ordinary differential equations can be solved by using Euler’s method. In another chapter we will discuss how Euler’s 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 0 , 3 . 1 2 = = + - y e y dx dy x in 0 ) 0 ( ), , ( y y y x f dx dy = = form. Solution ( 29 5 0 , 3 . 1 2 = = + - y e y dx dy x ( 29 5 0 , 2 3 . 1 = - = - y y e dx dy x In this case 08.02.1
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08.02.2 Chapter 08.02 ( 29 y e y x f x 2 3 . 1 , - = - Example 2 Rewrite ( 29 5 0 ), 3 sin( 2 2 2 = = + y x y x dx dy e y in 0 ) 0 ( ), , ( y y y x f dx dy = = form. Solution ( 29 5 0 ), 3 sin( 2 2 2 = = + y x y x dx dy e y ( 29 5 0 , ) 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 Euler’s method At 0 = x , we are given the value of . 0 y y = Let us call 0 = x as 0 x . Now since we know the slope of y with respect to x , that is, ( 29 y x f , , then at 0 x x = , the slope is ( 29 0 0 , y x f . Both 0 x and 0 y are known from the initial condition ( 29 0 0 y x y = . Figure 1 Graphical interpretation of the first step of Euler’s method. y Φ Step size, h x ) , ( 0 0 y x True value y 1 , Predicted value 1 x
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Euler’s Method 08.02.3 So the slope at 0 x x = as shown in Figure 1 is Slope Run Rise = 0 1 0 1 x x y y - - = ( 29 0 0 , y x f = From here ( 29 ( 29 0 1 0 0 0 1 , x x y x f y y - + = Calling 0 1 x x - the step size h , we get ( 29 h y x f y y 0 0 0 1 , + = (2) One can now use the value of 1 y (an approximate value of y at 1 x x = ) to calculate 2 y , and that would be the predicted value at 2 x , given by ( 29 h y x f y y 1
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