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# Problem2.3.2d - Problem 2.3.2(d The eigenvalue problem we...

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Problem 2.3.2 (d) The eigenvalue problem we need to address is (1) d 2 f dx 2 + l f = 0 BC1: f H 0 L = 0 BC2: d f dx H L L = 0 The general solution will depend on the sign of l : (i) l > 0 (2) f H x L = C 1 Cos I !!! l x M + C 2 Sin I !!! l x M Applying the BCs we find (3) f H 0 L = C 1 = 0 d f dx H L L = 0 = C 2 !!! l Cos I !!! l L M Thus (4) !!! l L = H n + 1 2 L p , n = 0, 1, 2... Thus for l > 0 the eigenfunctions are (5) f n H x L = Sin I !!! l x M = Sin @H n + 1 2 L p L x D (ii) For l = 0 we have a general solution (6) f H x L = C 1 x + C 2 Application of the BCs give (7) C 2 = 0, C 1 = 0 Thus l = 0 defines the trivial solution.
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