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Unformatted text preview: (3) Substituting Eq. (1) into Eq. (2) gives 3 or (4) where . Equation (4) is an ordinary 4 th order linear homogeneous differential equation with constant coefficients. According to the general theory, assume the solution to be of the form (5) Taking appropriate derivatives of Eq. (5) and substituting yields (6) Let Hence, (7) 4 Equation (7) shows a total of four distinct roots as expected since Eq. (4) is a 4 th order differential equation. Examination of Eq. (7) reveals that two of the roots are imaginary and the other two are real. Converting the imaginary value by multiplying with and The both are positive, real values. Then the general solution of Eq. (4) is (8) Since , and For nontrivial solution for A and C , the determinant for A and C must vanish. Hence, 5 (9) Since the sum of the terms in the parenthesis are nonzero, (10) . The smallest value of satisfying Eq. (10) is Hence, Substituting into above gives , Finally, 6 QED 7...
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 Spring '08
 MELVILLE

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