(a) From (30) and (31) of Section 6.4 we know that when n = 0, Legendre’s differential equation (1  x²)y'' - 2xy' = 0 has the polynomial solution y = P₀(x) = 1. Use (5) of Section 4.2 to show that a second Legendre function satisfying the DE for 1 < x < 1 is

(b) We also know from (30) and (31) of Section 6.4 that when n  1, Legendre’s differential equation (1  x²)y''  2x y' = 2y = 0 possesses the polynomial solution y = P₁(x) = x. Use (5) of Section 4.2 to show that a second Legendre function satisfying the DE for 1 < x < 1 is

(c) Use a graphing utility to graph the logarithmic Legendre functions given in parts (a) and (b).

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+ x 1-지 y = 2