MTH 122 — Calculus II
Essex County College — Division of Mathematics and Physics
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Lecture Notes #2 — Sakai Web Project Material
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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,
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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
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0
1
1
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Figure 2: Direction field of
y
0
=
x
+
y
, Euler’s fit, and actual solution.
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 Spring '10
 Ban
 Calculus, Division, Trigraph, Euler, direction ﬁeld

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