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527ass11 - 4 Concerning 18.3:6 we already did several parts...

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642:527 ASSIGNMENT 11 FALL 2009 Turn in starred problems Tuesday 12/01/2008. Section 17.7: 7 Section 17.8: 2 (a), (b)*, (d)*, 5 Section 18.3: 6 (i), (l), (m)*, 10 (a), (c)*, (f)*, (i) Comments, hints, instructions: 1. 17.7:7 shows that innocent looking but nonsepa- rated boundary conditions can lead to trouble. 2. 17.8:2(d): This is the Legendre equation that we studied earlier (Section 4.4). The requirement that the solution be bounded at x = 1 requires that it be one of the Legendre polynomials (see problem 3.A on Assignment 3); the boundary condition at x = 0 picks out some of these. (These two considerations together determine the eigenvalues.) The text solution for 17.8:2(e) may be helpful. 3. 17.8:5 is essentially the problem we encountered in studying the heat equation in a disk: a change of variables leads to Bessel’s equation.
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Unformatted text preview: 4. Concerning 18.3:6: we already did several parts of this problem in which the boundary conditions were homogeneous; the ones I have chosen here are inhomogeneous. You can use any method that you like, but I think that the clearest one is the method I outlined in class: Find the steady-state solution v ( x ) of the equation and boundary conditions (the book usually calls this u s ( x )), so that w ( x, t ) = u ( x, t )-v ( x ) will satisfy a homogeneous boundary value problem which you already know how to solve. 5. 18.3:10. Here you just have to ±nd the steady state solutions; part (c) is particularly illuminating (note that it is closely related to 18.3:6(i)). If you were asked to solve an initial value problem, you could use the steady-state to do so (method outlined in remark 3 above)....
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