*(a) Suppose that L = p(x) d2 dx2 + r(x) d dx + q(x). Consider b a vL(u) dx. By repeated
Chapter 5, Problem 5.5.11(choose chapter or problem)
*(a) Suppose that L = p(x) d2 dx2 + r(x) d dx + q(x). Consider b a vL(u) dx. By repeated integration by parts, determine the adjoint operator L such that b a [uL(v) vL(u)] dx = H(x) b a . What is H(x)? Under what conditions does L = L, the self-adjoint case? [Hint: Show that L = p d2 dx2 + 2 dp dx r d dx + d2p dx2 dr dx + q . (b) If u(0) = 0 and du dx(L) + u(L)=0, what boundary conditions should v(x) satisfy for H(x)| L 0 = 0, called the adjoint boundary conditions? 5.
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