Legendres differential equation reads where is some (non-negative) real number. (a) Assume a power series solution,

Legendre’s differential equation reads

where ℓ is some (non-negative) real number.
(a) Assume a power series solution, and obtain a recursion relation for the constants

(b) Argue that unless the series truncates (which can only happen if ℓ is an integer), the solution will diverge at x = 1.

(c) When ℓ is an integer, the series for one of the two linearly independent solutions (either or depending on whether ℓ is even or odd) will truncate, and those solutions are called Legendre polynomials Pℓ(x).
Find P₃(x), P₃(x), P₃(x), and P₃(x) from the recursion relation. Leave your answer in terms of either a₀ or a₁.

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(1-x²) ²2 - 2x + (+1) ƒ = 0, 52 l f df dx (2.208)