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Unformatted text preview: Math 306 – Lab 2 Improved Euler’s Method Printed Name: Please carefully work all of the following problem(s). You must SHOW YOUR WORK to receive ANY credit! You can work together, but must write up your own solutions. You must include your commented Mathematica code! Read Section 2.5. 1 Improved Euler’s Method This lab will investigate the Improved Euler’s Method for approximating the solutions to differential equations. As an example, the simple differential equation dy dx = e- x 2 , y (0) = 1 . has a unique solution – namely y ( x ) = 1+ Z x e- u 2 du . Unfortunately, there is no elementary way to perform this integration. Instead, we shall try to approximate the solution. Consider the general IVP dy dx = f ( x,y ) , y ( x 1 ) = y 1 . Euler’s Method assumed that the slope is constant over the interval [ x n ,x n + h ]. This is usually not the case. So, we seek to improve Euler’s Method with the following modifications. See Figure 1. The blue curve is the true solution, y ( x ). The dark green dotted line represent the tangent line through the point ( x n ,y ( x n )). The dark red dotted line represents the slope of the line tangent to the intersection of the previous (green-dotted) tangent line and the vertical line through the point x n +1 . Finally, the dark blue dash-dotted line represents the average of the two slopes. The intersection of the dash-dot line with the vertical line through x n +1 is the point we need to use....
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This note was uploaded on 01/13/2012 for the course MATH 306 taught by Professor Keithemmert during the Spring '11 term at Tarleton.
- Spring '11