1. Series solution to second order ODE I Consider the ODE given by xy00(x) + (1 − x)y 0 (x) + λy(x) = 0 for λ ∈ R. (1) Show that x = 0 is a regular singular point. (2) Find the indicial equation by the Frobenius method (3) Find the roots of the indicial equation (4) Write down the recursion relation for the coefficents hj and λ ∈ R (5) Consider now the case that λ ∈ N0 and set n = λ. Moreover, we assume h0 = 1. Show that in this case hj = 0 for all j > m. (6) Find a closed form solution Ln for n ∈ N. (7) Compute the polynomials L0,L1, . . . ,L4. 2. Series solution to second order ODE II Consider the ODE given by (1 − x 2 )y 00(x) − xy0 (x) + λ 2 y(x) = 0 where λ ∈ R. (1) Show that x = 0 is an ordinary point (2) Use the power series method with the ansatz h(x) = P∞ j=0 hjx j to find a recursion relation for the coefficents. (3) Consider the case λ ∈ N and set λ = p. Show that if p is even, the series (we name the solution Fp with h0 = 1 and h1 = 0 terminates after finitely many steps. Moreover, show that if p is odee, the series (we name the solution Gp) with h0 = 0 and h1 = 1 terminates after finitely many steps. (4) Define the polynomials Tp Tp(x) = ( (−1) p 2 Fp(x) p is even, (−1) p−1 2 pG(x) p is odd. and compute T0, T1, T2. (5) Please show that Tp(x) = cos(n arccos x) for p = 0, 1, 2. 1 2 3. Series solution to second order ODE III Consider the ODE given by x 2 y 00(x) + 2xy0 (x) + (x 2 − n(n + 1))y(x) = 0 for n ∈ N. Please answer the following questions (1) Show that x = 0 is a regular singular point (2) Find the indicial equation by the Frobenius method (3) Find the roots of the indicial equation (4) Use the root α1 with α1 > α2 to write down the recursion relation for the coefficents hj for j ≥ 1. (5) Write down the recursion relation for the coefficents dk = h2k under the assumption that k = 0. (6) Solve the recursion relation to show that y0(x) = sin x x is a solution to the above recursion for α = 0. 4. Properties of Bessel functions We consider for n ∈ N the Bessel functions Jn defined by the power series Jn(x) = X∞ j=0 (−1)k k!(k + n)! x 2 2j+n Please answer the following questions: (1) Compute for each n ∈ N the convergence radius ρ of the series. (2) Show that J0(0) = 1 and Jn(0) = 0 for all n ≥ 1. (3) Show that for all n ≥ 1 we have the derivative identity d dx (x nJn(x)) = x nJn−1(x)