hw9_soln - 5.3.8 Multiplying the equation by φ yields: 0 =...

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Unformatted text preview: 5.3.8 Multiplying the equation by φ yields: 0 = φ d 2 φ dx 2 + ( λ- x 2 ) φ 2 = d dx φ dφ dx- dφ dx 2 + λφ 2- x 2 φ 2 . Integrating this identity over the interval 0 ≤ x ≤ 1 allows us to conclude: φ dφ dx 1- Z 1 " dφ dx 2 + x 2 φ 2 # dx =- λ Z 1 φ 2 dx. If φ is an eigenfunction, and hence not identically zero, we can now divide by- R 1 φ 2 dx to obtain the standard Rayleigh quotient λ =- φ dφ dx 1 + R 1 h ( dφ/dx ) 2 + x 2 φ 2 i dx R 1 φ 2 dx The first derivatives of φ vanish at the boundary which means λ = R 1 h ( dφ/dx ) 2 + x 2 φ 2 i dx R 1 φ 2 dx ≥ . If λ = 0, we would then have 0 = Z 1 dφ dx 2 dx + Z 1 x 2 φ 2 dx ≥ since each term in the integrand is nonnegative. This implies that each integral vanishes, and that each integrand vanishes. In particular, x 2 φ 2 ( x ) = 0 for all 0 ≤ x ≤ 1, which implies that φ ( x ) = 0 for all < x ≤ 1 and hence φ (0) = 0 by continuity. Thus φ is the trivial function and hence λ = 0 cannot be an eigenvalue.an eigenvalue....
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hw9_soln - 5.3.8 Multiplying the equation by φ yields: 0 =...

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