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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 ) + λφ = 0 1 < x < 2 φ (1) = 0 φ (2) = 0 x 2 φ + 2 + λφ = 0 An Euler Eq. (29.1) Let φ ( x ) = x r r ( r 1) + 2 r + λ = r 2 + r + λ = 0 . (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 = 0 φ (2) = c 2 2 1 2 log 2 = 0 c 2 = 0 (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 = 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 ) = 0 (29.8) 2 r 1 r 2 = 1 ρ ( r 1 r 2 ) ln 2 = 1 = ρ 2 πin ( r 1 r 2 ) ln 2 = 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 . For λ > 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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