Problem5.3.9c - Problem 5.3.9 (c) If the equation is...

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Problem 5.3.9 (c) If the equation is equidimensional then it will have solutions of the form f H x L ~ x p . This implies then that (by substitution) (1) p H p - 1 L + p + l = 0, ï p 2 = -l If l > 0, then p = i è!!! l . thus the solution is of the form (2) f H x L = c 1 x i è!!! l + c 2 x - i è!!! l Note that x = e ln H x L . Thus we can also write the general solution as (3) f H x L = c 1 e i è!!! l ln H x L + c 2 e - i è!!! l ln H x L But (4) e i a = Cos H a L + i Sin H a L from Euler's formula. We can therefore write the solution as (5) f H x L = c 3 Cos A è!!! l ln H x LE + c 4 Sin A è!!! l ln H x LE The boundary condition f H 1 L = 0 implies that c 3 = 0, whereas f H b L = 0, gives (6) Sin A è!!! l ln H b LE = 0, or è!!! l ln H b L = n p , n = 1, 2, .. Thus we have the eigenvalues
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