Chapter 6
Series Solutions of Linear Differential Equations
The methods of Chapter 4 give exact formulas, but only for special equations: constant coefficients or the Cauchy–Euler form. As soon as the coefficients depend on in a more general way, as in or in Bessel's equation, those methods stop working, and in most cases no solution can be written with elementary functions at all. This chapter gives a method that works anyway: we build the solution as an infinite series and let the differential equation produce its coefficients one by one.
The chapter has three sections:
- 6.1 Ordinary points: a review of power series, a careful treatment of index shifting, the theorem that guarantees power series solutions and tells you their radius of convergence, and recurrence relations.
- 6.2 Regular singular points: the Frobenius method, the indicial equation, and the three cases that decide whether the second solution contains a logarithm.
- 6.3 Bessel's equation: the gamma function, Bessel functions of the first and second kind, the half-integer case in elementary form, recurrence identities and zeros, plus a short look at Legendre polynomials.
What you will be able to do
- Shift summation indices confidently and combine series into a single sum.
- Classify points as ordinary, regular singular or irregular singular, and predict the radius of convergence of series solutions.
- Derive a recurrence relation and compute series solutions, including initial-value problems and expansions about .
- Solve the indicial equation and construct Frobenius solutions in all three cases.
- Work with , and the gamma function, as needed in cylindrical problems such as vibrating membranes, heat conduction and waveguides.
Prerequisites
Taylor series and the ratio test from calculus, the basic theory of linear equations (Section 4.1, in particular reduction of order) and the Cauchy–Euler equation (Section 4.5), whose indicial equation reappears in Section 6.2.