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mth.122.handout.02

# mth.122.handout.02 - MTH 122 Calculus II Essex County...

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MTH 122 — Calculus II Essex County College — Division of Mathematics and Physics 1 Lecture Notes #2 — Sakai Web Project Material 1 Euler’s Method Direction fields, as I have been drawing them, are best done on a computer. They’re 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, let’s say I don’t know the solution, but I still want to approximate the graph of the solution over the direction field. I’m 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 Euler’s method, and that’s what I will concentrate on here. Let’s look at an example, y 0 = x + y, y (0) = 1 , where I will have Grapher plot the direction field , and a numerical solution, through the point (0 , 1), using Euler’s method with a step size of 0 . 1. 5 -2.5 0 2.5 5 -2.5 2.5 Figure 1: Direction field for y 0 = x + y , and a numerical solution that contains (0 , 1). Fact is, Euler’s 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 it’s 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 Euler’s method is an approximate visualization, so let’s now look at how 1

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-1 0 1 1 2 Figure 2: Direction field of y 0 = x + y , Euler’s fit, and actual solution.
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mth.122.handout.02 - MTH 122 Calculus II Essex County...

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