Mathematical Methods for Physics and Engineering - Matematica.NET

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONScoefficient of the highest power z N ; such a power now exists because of ourassumed form of solution.◮By assuming a polynomial solution find the values of λ in (16.34) for which such a solutionexists.We assume a polynomial solution to (16.34) of the form y = ∑ Nn=0 a nz n . Substituting thisform into (16.34) we findN∑ [n(n − 1)an z n−2 − 2zna n z n−1 + λa n z n] =0.n=0Now, instead of starting with the lowest power of z, we start with the highest. Thus,demanding that the coefficient of z N vanishes, we require −2N + λ =0,i.e.λ =2N, aswefound in the previous example. By demanding that the coefficient of a general power of zis zero, the same recurrence relation as above may be derived and the solutions found. ◭16.6 Exercises16.1 Find two power series solutions about z = 0 of the differential equation(1 − z 2 )y ′′ − 3zy ′ + λy =0.Deduce that the value of λ for which the corresponding power series becomes anNth-degree polynomial U N (z) isN(N + 2). Construct U 2 (z) andU 3 (z).16.2 Find solutions, as power series in z, of the equation4zy ′′ +2(1− z)y ′ − y =0.Identify one of the solutions and verify it by direct substitution.16.3 Find power series solutions in z of the differential equationzy ′′ − 2y ′ +9z 5 y =0.Identify closed forms for the two series, calculate their Wronskian, and verifythat they are linearly independent. Compare the Wronskian with that calculatedfrom the differential equation.16.4 Change the independent variable in the equationd 2 f df+2(z − a) +4f =0dz2 dz (∗)from z to x = z − α, and find two independent series solutions, expanded aboutx = 0, of the resulting equation. Deduce that the general solution of (∗) is∞∑ (−4) m m!f(z,α) =A(z − α)e −(z−α)2 + B(z − α) 2m ,(2m)!m=0with A and B arbitrary constants.16.5 Investigate solutions of Legendre’s equation at one of its singular points asfollows.(a) Verify that z = 1 is a regular singular point of Legendre’s equation and thatthe indicial equation for a series solution in powers of (z − 1) has roots 0and 3.(b) Obtain the corresponding recurrence relation and show that σ = 0 does notgive a valid series solution.550