Mathematical Methods for Physics and Engineering - Matematica.NET

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS(b) Find one series solution in powers of z. Give a formal expression for asecond linearly independent solution.(c) Deduce the values of λ for which there is a polynomial solution P N (z) ofdegree N. Evaluate the first four polynomials, normalised in such a way thatP N (0) = 1.16.11 Find the general power series solution about z =0oftheequationz d2 y dy+(2z − 3)dz2 dz + 4 z y =0.16.12 Find the radius of convergence of a series solution about the origin for theequation (z 2 + az + b)y ′′ +2y = 0 in the following cases:(a) a =5,b=6; (b)a =5,b=7.Show that if a and b are real and 4b >a 2 , then the radius of convergence isalways given by b 1/2 .16.13 For the equation y ′′ + z −3 y = 0, show that the origin becomes a regular singularpoint if the independent variable is changed from z to x =1/z. Hence find aseries solution of the form y 1 (z) = ∑ ∞0 a nz −n .Bysettingy 2 (z) =u(z)y 1 (z) andexpanding the resulting expression for du/dz in powers of z −1 , show that y 2 (z)has the asymptotic form[( )] ln zy 2 (z) =c z +lnz − 1 +O ,2zwhere c is an arbitrary constant.16.14 Prove that the Laguerre equation,z d2 y dy+(1− z) + λy =0,dz2 dzhas polynomial solutions L N (z) ifλ is a non-negative integer N, and determinethe recurrence relationship for the polynomial coefficients. Hence show that anexpression for L N (z), normalised in such a way that L N (0) = N!, isN∑ (−1) n (N!) 2L N (z) =(N − n)!(n!) 2 zn .n=0Evaluate L 3 (z) explicitly.16.15 The origin is an ordinary point of the Chebyshev equation,(1 − z 2 )y ′′ − zy ′ + m 2 y =0,which therefore has series solutions of the form z ∑ σ ∞0 a nz n for σ =0andσ =1.(a) Find the recurrence relationships for the a n in the two cases and show thatthere exist polynomial solutions T m (z):(i) for σ =0,whenm is an even integer, the polynomial having 1 (m +2)2terms;(ii) for σ =1,whenm is an odd integer, the polynomial having 1 (m +1)2terms.(b) T m (z) is normalised so as to have T m (1) = 1. Find explicit forms for T m (z)for m =0, 1, 2, 3.552