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6.6 Euler’s Method
Leonhard Euler
1707  1783
Leonhard Euler made a
huge number of
contributions to
mathematics, almost half
after he was totally blind.
(When this portrait was
made he had already lost
most of the sight in his right
eye.)
Greg Kelly, Hanford High School, Richland, Washington
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View Full Document Leonhard Euler
1707  1783
It was Euler who originated
the following notations:
e
(base of natural log)
( 29
f x
(function notation)
π
(pi)
i
( 29
1

(summation)
∑
y
∆
(finite change)
→
There are many differential equations that can not be solved.
We can still find an approximate solution.
We will practice with an easy one that can be solved.
2
dy
x
dx
=
Initial value:
0
1
y
=
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dy
x
dx
=
0
1
y
=
n
n
x
n
y
dy
dx
dy
1
n
y
+
0.5
dx
=
0
0
1
0
0
1
1
.5
1
1
.5
1.5
2
1
1.5
2
1
2.5
dy
dx
dy
dx
⋅
=
1
n
n
y
dy
y
+
+
=
→
3
1.5
2.5
3
1.5
4.0
4
2.0
4.0
dy
dx
dy
dx
⋅
=
1
n
n
y
dy
y
+
+
=
2
dy
x
dx
=
n
n
x
n
y
dy
dx
dy
1
n
y
+
0.5
dx
=
0
0
1
0
0
1
1
.5
1
1
.5
1.5
2
1
1.5
2
1
2.5
0
1
y
=
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dy
x
dx
=
( 29
0,1
0.5
dx
=
2
dy
x dx
=
2
y
x
C
=
+
1
0
C
= +
2
1
y
x
=
+
Exact Solution:
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This note was uploaded on 03/10/2008 for the course MATH 116 taught by Professor Chale during the Fall '08 term at Cal Poly Pomona.
 Fall '08
 chale
 Math, Differential Calculus

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