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Chapter 12: Q7-6P (page 562)
Orthogonal functions are members of a function space, which is a bilinear vector space.
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For Problems 1 to 4 , find one (simple) solution of each differential equation by series, and then find the second solution by the "reduction of order" method, Chapter 8 , Section 7 (e).
To show that .
Solve the differential equations in Problems 5 to 10 by the Frobenius method; observe that you get only one solution. (Note, also, that the two values of are equal or differ by an integer, and in the latter case the larger gives the one solution.) Show that the conditions of Fuchs's theorem are satisfied. Knowing that the second solution is X times the solution you have, plus another Frobenius series, find the second solution.
Verify the recursion relationsas follows:
a) DifferentiateWith respect toto get equate coefficients ofrole="math" localid="1654857725406"
b) Differentiate with respect to to get equate coefficients of
c) Combine (a) and (b) to get . Substitute the series for and equate coefficients of
Use problem 7 to show that
Plm (x) = (-1)m (l+m)!/(l-m)! (1-x2)/2l! dl-m/dxl-m(x2-1)l
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