EGM6321 - Principles of Engineering Analysis 1, Fall 2010
Mtg 29: Tue, 26 Oct10
HW5.5 p.28-3 : continued
2.8) Now solve L2_ODE_CC with Gaussian distribution
Coefficients: : Find these such that L2_ODE_CC Eq.(2)p.27-2 accept as characteristic equation:
2.8.1)
2.8.2) END HW5.5
Variation of Parameters: (continued)
Instead of doing the homogeneous ( F09 and King ) cases Eq.(1)p.28-4 , Let's consider directly the non-homogeneous case, then recover the homogeneous as a particular case. So consider:
Given: homogeneous solution
Find: 2nd homogeneous solution
particular solution
Method
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(1)
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Where is unknown and is known
Add the following equations:
, where
After summing: L2_ODE_VC with missing dependeant variable
Where , thus is a homogeneous solution.
Order reduction method :
Where and
Solve for given by Eq.(6)p.10-3 with given by Eq.(1)p.10-3 .
Eq.(1)p.10-3 :
Where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle \left [ \int_{}^{} \left \{ a_1(x)\frac{2u_1'(x)}{u_1(x)} \right \}\,dx \right ] = \int_{}^{} a_1(x)\,dx+2 \log {u_1(x)} \ }
and
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(1)
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From Eq.(6)p.10-3: , where and
From Eq.(1)p.29-2:
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(2)
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Where
References