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Unformatted text preview: Math 306 Lab 2 Improved Eulers 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 Eulers Method This lab will investigate the Improved Eulers 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 . Eulers 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 Eulers 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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- Spring '11