EGM6321 - Principles of Engineering Analysis 1, Fall 2009
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Mtg 17: Thur, 10Oct09
Linearity
superposition
Homogeneous Solution
- Euler Equations
- Trial solution (undefined coefficient)
- Reduction of order method 2: Undetermined factor
Homogeneous L2_ODE_VC: cf. Eq.(1) P.3-1
|
(1)
|
Where
can be substituted for
or
in Eq(1)
Given one homogeneous solution
known
Find second homogeneous solution
such that
|
(3)
|
Where
are constants
Assume full homogeneous solution
|
(2)
|
Where
is an unknown to be determined
Where
is known
"Full" = includes
Add the following:
and
and
To get
by Eq(1) p17-1
Reduce to
Since
is a homogeneous solution
, NOTE missing dependent variable U in front of
term
Let
homogeneous L1_ODE_VC for Z
|
(1)
|
Solve for Z,
- integration factorial method (HW)
- Direct integration (because Eq(1) is homogeneous)
Where
are known
Integrate
, where k is a constant
|
(2)
|
where
where
Homogeneous solution
|
(1)
|
where
and
|
(2)
|
HW: obtain Eq.(2) P.17-3
using the integrating factor method