M257-316Notes_Lecture33

M257-316Notes_Lecture33 - Chapter 29 Lecture 33 Variable...

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Chapter 29 Lecture 33 Variable coefficient BVP - eigenfunctions involving solutions to the Euler Equation: Example 29.1 Eigenfunctions involving solutions to an Euler Equation: ( x 2 φ 0 ) 0 + λφ =0 1 <x< 2 φ (1) = 0 φ (2) = 0 x 2 φ 0 +2 0 + λφ = 0 An Euler Eq. (29.1) Let φ ( x )= x r r ( r 1)+2 r + λ = r 2 + r + λ . (29.2) Therefore r = 1 ± 1 4 λ 2 = r 1 ,r 2 . (29.3) 29.1 Cases: λ = 1 4 : φ ( x c 1 x 1 2 + c 2 x 1 2 log x (29.4) φ (1) = c 1 φ (2) = c 2 2 1 2 log 2 = 0 c 2 (29.5) 167
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Lecture 33 Variable coefficient BVP - eigenfunctions involving solutions to the Euler Equation: so there is no Eigenfunction for λ =1 / 4. λ 6 = 1 4 : φ ( x )= c 1 x r 1 + c 2 x r 2 (29.6) φ (1) = c 1 + c 2 =0 c 2 = c 1 (29.7) φ (2) = c 1 ( 2 r 1 2 r 2 ) (29.8) 2 r 1 r 2 ρ ( r 1 r 2 )ln2 =1= ρ 2 πin ( r 1 r 2 )ln2=2 πin r 1 r 2 = 1 4 λ =2 πni/ ln(2) (29.9) Thus to obtain nontrivial solutions we require 1 4 λ< 0 λ> 1 4 .F o r 1 4 1 4 λ = i 4 λ 1=2 πni/ ln(2) . (29.10) The Eigenvalues are: λ n = 1 4 + π 2 n 2 [ln(2)] 2 , 4 λ n 1= 4 π 2 n 2 [ln(2)] 2 =(2 β n ) 2 β n =( nπ/ ln 2) . (29.11)
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This note was uploaded on 05/04/2011 for the course MATH 25 taught by Professor Lo during the Spring '11 term at BC.

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M257-316Notes_Lecture33 - Chapter 29 Lecture 33 Variable...

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