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EulersMethodLab - Math 306 Lab 1 Eulers Method Printed Name...

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Math 306 – Lab 1 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 2.4. 1 Euler’s Method This lab will investigate 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 0 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 . The idea is We pick an initial point, say ( x 1 , y 1 ) The slope through the point ( x 1 , y 1 ) is given by f ( x 1 , y 1 ). We move a small distance along the slope segment through this point to the next point ( x 2 , y 2 ). Note that y 2 is an approximation to y ( x 2 ). At the point ( x 2 , y 2 ), we have a new slope given by f ( x 2 , y 2 ). We now move a small distance along the slope segment through ( x 2 , y 2 ) to the next point ( x 3 , y 3 ). Again, y 3 y ( x 3 ). Etc. A more mathematical version of this process is: Choose a step size, say h > 0. Your first point is ( x 1 , y 1 ). Your next point is ( x 2 , y 2 ), where x 2 = x 1 + h and y 2 = y 1 + h · f ( x 1 , y 1 ). Your next point is ( x 3 , y 3 ), where x 3 = x 2 + h and y 3 = y 2 + h · f ( x 2 , y 2 ). Etc. Your n + 1 st point is ( x n 1 , y n 1 ), where x n 1 = x n + h and y n 1 = y n + h · f ( x n , y n ). Hence, the numbers y 1 , y 2 , . . . , y n , . . . form an approximate solution to y ( x ). Now, return to the IVP dy dx = e - x 2 y (0) = 1 . Choose a step size: h = 1. ( x 1 , y 1 ) = (0 , 1). The slope through this point is f (0 , 1) = 1. So... x 2 = x 1 + h = 0 + 1 = 1 y 2 = y 1 + h · f ( x 1 , y 1 ) = 1 + 1 · 1 = 2 . Hence, ( x 2 , y 2 ) = (1 , 2).
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