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Unformatted text preview: MTH 122 Calculus II Essex County College Division of Mathematics and Physics 1 Lecture Notes #2 Sakai Web Project Material 1 Eulers Method Direction fields, as I have been drawing them, are best done on a computer. Theyre really tedious when done by hand, but once drawn they can give you a fairly good idea of what solutions may look like. In the prior examples I actually knew the solutions, so I was able to graph those superimposed over the direction field. Now, lets say I dont know the solution, but I still want to approximate the graph of the solution over the direction field. Im using a program called Grapher, 2 and it allows me to draw both the direction field for a given differential equation, and several types of approximate solutions. One simple technique that the book mentions is Eulers method, and thats what I will concentrate on here. Lets look at an example, y = x + y, y (0) = 1 , where I will have Grapher plot the direction field , and a numerical solution, through the point (0 , 1), using Eulers method with a step size of 0 . 1.-5-2.5 2.5 5-2.5 2.5 Figure 1: Direction field for y = x + y , and a numerical solution that contains (0 , 1). Fact is, Eulers method is just a visual, and I doubt seriously that anyone would be able to tell me the exact solution even after looking at the above image. However, I did the nasty work of finding the solution, and its y = 2 e x- x- 1 . You should be able to verify that this is a solution and satisfies the initial condition. But I really want to emphasize that Eulers method is an approximate visualization, so lets now look at how 1-1 1 1 2 Figure 2: Direction field of y = x + y , Eulers fit, and actual solution....
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