Consider the unusual eigenvalue problem vxx = v for 0 < x

Chapter 4, Problem 4.3.12

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Consider the unusual eigenvalue problem vxx = v for 0 < x < l vx(0)= vx(l)= v(l)v(0) l . (a) Show that =0 is a double eigenvalue. (b) Get an equation for the positive eigenvalues >0. (c) Letting = 1 2l, reduce the equation in part (b) to the equation sin cos =sin2 . (d) Use part (c) to nd half of the eigenvalues explicitly and half of them graphically. (e) Assuming that all the eigenvalues are nonnegative, make a list of all the eigenfunctions. (f) Solve the problem ut =kuxx for0 < x < l, with the BCs given above, and with u(x, 0)= (x). (g) Show that, as t , limu(x, t)= A+ Bx for some constantsA , B, assuming that you can take limits term by term.

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