Consider a fourth-order linear differential operator, L = d4 dx4 . (a) Show that uL(v)
Chapter 5, Problem 5.5.8(choose chapter or problem)
Consider a fourth-order linear differential operator, L = d4 dx4 . (a) Show that uL(v) vL(u) is an exact differential. (b) Evaluate 1 0 [uL(v) vL(u)] dx in terms of the boundary data for any functions u and v. (c) Show that 1 0 [uL(v) vL(u)] dx = 0 if u and v are any two functions satisfying the boundary conditions (0) = 0, (1) = 0 d dx (0) = 0, d2 dx2 (1) = 0. (d) Give another example of boundary conditions such that 1 0 [uL(v) vL(u)] dx = 0. (e) For the eigenvalue problem [using the boundary conditions in part (c)] d4 dx4 + ex = 0, show that the eigenfunctions corresponding to different eigenvalues are orthogonal. What is the weighting function?
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