Mathematical Methods for Physics and Engineering : A Comprehensive Guide

16 Series solutions of ordinary differential equations In the previous chapter the solution of both homogeneous and non-homogene ...

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS wherey 1 (x)andy 2 (x)arelinearly independentsolutions of (16.1), andc 1 and ...

16.1 SECOND-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS can be written as the sum of the solution to the homogeneous equationyc ...

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS singular points, whereas any singular point not satisfying both these criter ...

16.2 SERIES SOLUTIONS ABOUT AN ORDINARY POINT Equation Regular Essential singularities singularities Hypergeometric z(1−z)y′′+[c ...

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS power series of the form (see section 24.11) y(z)= ∑∞ n=0 anzn. (16.9) Moreo ...

16.2 SERIES SOLUTIONS ABOUT AN ORDINARY POINT agivena 0. Alternatively, starting witha 1 , we obtain the odd coefficientsa 3 ,a ...

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS By demanding that the coefficients of each power ofzvanish separately, we ob ...

16.3 SERIES SOLUTIONS ABOUT A REGULAR SINGULAR POINT then at least one ofp(z)andq(z) is not analytic atz= 0, and in general we s ...

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS Settingz= 0, all terms in the sum withn>0 vanish, implying that [σ(σ−1) + ...

16.3 SERIES SOLUTIONS ABOUT A REGULAR SINGULAR POINT
Find the power series solutions aboutz=0of 4 zy′′+2y′+y=0. Dividing throug ...

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS We may check thaty 1 (z)andy 2 (z) are indeed linearly independent by comput ...

16.3 SERIES SOLUTIONS ABOUT A REGULAR SINGULAR POINT one solution in the form of a Frobenius series. We therefore substitutey=zσ ...

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS is required, i.e.z= 0, is in fact an ordinary point of the ODE rather than a ...

16.4 OBTAINING A SECOND SOLUTION to (16.21) isy 1 =z/(1−z)^2. Substituting forpandy 1 in (16.25) we have y 2 (z)= z (1−z)^2 ∫z ( ...

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS of the indicial equation areσ=σ 1 andσ=σ 2 then it follows that Ly(z, σ)=a 0 ...

16.4 OBTAINING A SECOND SOLUTION which is equal to zero ifσ=σ 2. As previously, since∂/∂σandLare operators that differentiate wi ...

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS derivative method. Let us first consider the case where the two solutions of ...

16.5 POLYNOMIAL SOLUTIONS is a positive integer or zero, then we are left with a finite polynomial of degree N′=N+σas a solution ...

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS coefficient of the highest powerzN; such a power now exists because of our a ...