Begin with the general solution in (37) of y.4/ 4y D 0.
Chapter , Problem 19(choose chapter or problem)
Begin with the general solution in (37) of y.4/ 4y D 0. First note that y.0/ D y0 .0/ D 0 implies that C D A and D D B. Then impose the conditions y.L/ D y0 .L/ D 0 to get two homogeneous linear equations in A and B. Hence the determinant of coefficients of A and B must vanish; deduce from this that cosh L cos L D 1. Conclude that the nth eigenvalue is n D 4 n=L4 where fng 1 1 are the positive roots of the equation cosh x cos x D 1 (see Fig. 10.1.3). Finally, show that an associated eigenfunction is yn.x/ D .sinh n sin n/ cosh nx L cos nx L .cosh n cos n/ sinh nx L sin nx L : 1
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