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08.02.1
Chapter 08.02
Euler’s Method for Ordinary Differential Equations
After reading this chapter, you should be able to
:
1.
develop Euler’s Method for solving ordinary differential equations,
2.
determine how the step size affects the accuracy of a solution,
3.
derive Euler’s formula from Taylor series, and
4.
use Euler’s method to find approximate values of integrals.
What is Euler’s method?
Euler’s method is a numerical technique to solve ordinary differential equations of the form
( ) ( )
0
0
,
,
y
y
y
x
f
dx
dy
=
=
(1)
So only first order ordinary differential equations can be solved by using Euler’s method.
In
another chapter we will discuss how Euler’s method is used to solve higher order ordinary
differential equations or coupled (simultaneous) differential equations.
How does one write a
first order differential equation in the above form?
Example 1
Rewrite
( )
5
0
,
3
.
1
2
=
=
+
−
y
e
y
dx
dy
x
in
0
)
0
(
),
,
(
y
y
y
x
f
dx
dy
=
=
form.
Solution
( )
5
0
,
3
.
1
2
=
=
+
−
y
e
y
dx
dy
x
( )
5
0
,
2
3
.
1
=
−
=
−
y
y
e
dx
dy
x
In this case

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