Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.
••• (Section 8.6) The study (either theoretical or numerical) of the θ equation (8.65) for ϴ(θ) is made difficult because the differential equation has singularities at θ = 0 and π. (The sin θ in the denominator makes the first term infinite at θ = 0 and π. Indeed this is exactly why there are no acceptable solutions for most values of l.) Nevertheless, if you have access to software that can solve differential equations numerically, you can get some insight to the acceptable and unacceptable solutions. (a) Write down the differential equation (8.65) for m = 0 and 1=2. Solve it numerically for the boundary conditions ϴ(π/2) = 1 and ϴ′(π/2) = 0. Plot your result for 0 ≤ θ ≤ π, and note that this solution looks perfectly acceptable. Repeat with the boundary conditions ϴ(π/2) = 0 and ϴ′(π/2) = 1 and explain why this solution appears to be unacceptable. (b) Repeat part (a) with m = 0 but l = 1.75. Explain why both solutions appear to be unacceptable; that is, there is no acceptable solution for these values of m and l.
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