Mathematical Methods for Physics and Engineering - Matematica.NET

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONSintegrating factor. This is given, as in equation (17.39), by[ ∫ xc − (a + b +1)u − 1+2uF(x) =expu(1 − u)[ ∫ x]c − 1 − (a + b − 1)u=expduu(1 − u)[ ∫ x( c − 1=exp1 − u + c − 1u]du− a + b − 1 )]du1 − u=exp[(a + b − c)ln(1− x)+(c − 1) ln x ]= x c−1 (1 − x) a+b−c .When the equation is multiplied through by F(x) ittakestheform[x c (1 − x) a+b−c+1 y ′ ] ′− abx c−1 (1 − x) a+b−c y =0.Now, for the corresponding Sturm–Liouville operator to be Hermitian, the conditionsto be imposed are as follows.(i) The boundary condition (17.37); if c>0anda + b − c +1> 0, this is satisfiedautomatically for 0 ≤ x ≤ 1, which is thus the natural interval in this case.(ii) The weight function x c−1 (1 − x) a+b−c must be finite and not change sign in theinterval 0 ≤ x ≤ 1. This means that both exponents in it must be positive, i.e.c − 1 > 0anda + b − c>0.Putting together the conditions on the parameters gives the double inequality a + b>c>1. ◭Finally, we consider Bessel’s equation,x 2 y ′′ + xy ′ +(x 2 − ν 2 )y =0,which may be converted into Sturm–Liouville form, but only in a somewhatunorthodox fashion. It is conventional first to divide the Bessel equation by xand then to change variables to ¯x = x/α. In this case, it becomes¯xy ′′ (α¯x)+y ′ (α¯x) − ν2¯x y(α¯x)+α2¯xy(α¯x) =0, (17.41)where a prime now indicates differentiation with respect to ¯x. Dropping the barson the independent variable, we thus have[xy ′ (αx)] ′ − ν2x y(αx)+α2 xy(αx) =0, (17.42)whichisinSLformwithp(x) =x, q(x) =−ν 2 /x, ρ(x) =x and λ = α 2 .Itshould be noted, however, that in this case the eigenvalue (actually its squareroot) appears in the argument of the dependent variable.568