Question

4. (a) Solve the differential equation (1 − x 2 )y ′′ − 2xy′ + λ(λ + 1)y = 0 using power series centered at 0 , in which λ is a constant. Write your solution as a linear combination of two independent solutions whose coefficients are expressed in terms of λ . Compute the coefficients of each solution up to and including the x 5 term. Without computing them, what is the smallest possible value of the radius of convergence of each solution and why?

(b) When λ = N is a nonnegative integer, show that one of the solutions obtained above is a polynomial PN of degree N . Find P0(x), P1(x), P2(x), P3(x) .

Answer 1

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