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Problem5.3.4b - Problem 5.3.4(b We assume the form for the...

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Problem 5.3.4(b) We assume the form for the solution (1) u H x, t L = f H x L h H t L Substituting into the PDE gives (2) 1 ÅÅÅÅ h h ÅÅÅÅÅÅÅ t = 1 ÅÅÅÅ f i k j j k 2 f ÅÅÅÅÅÅÅÅÅ x 2 - V 0 „f ÅÅÅÅÅÅÅ x y { z z = -l 2 The appropriate eigenvalue problem is (3) k 2 f ÅÅÅÅÅÅÅÅÅ x 2 - V 0 „f ÅÅÅÅÅÅÅ x + l 2 f = 0, f H 0 L = 0, f H L L = 0 To find the general solution we let f H x L = e a x . Substituting into the ODE gives (4) H k a 2 - V 0 a + l 2 L e a x = 0, or (5) k a 2 - V 0 a + l 2 = 0 Since we need to have oscillatory solutions, the eigenvalues must be complex. Thus (6) a = V 0 "##################### V 0 2 - 4 l 2 k ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ 2 k = V 0 Â "################ ###### 4 l 2 k - V 0 2 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 2 k Thus the general solution is (7) f H x L = c 1 e a 1 x + c 2 e a 2 x where (8) a 1 = V 0 + Â "##################### 4 l 2 k - V 0 2 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 2 k , a

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