Chapter 6 Series Solutions of Linear Equations
- 6.1 Solutions About Ordinary Points
- 6.2 Solutions About Singular Points
- 6.3 Two Special Equations
- Series Solutions of Linear Equations About a Singular Point (Method of Froebenius)
Series Solutions of Linear Equations About an Ordinary Point
Procedure for finding a general solution, in terms of a Power Series, of a second-order linear DE, around the Ordinary Point x0=0.
- Put the DE in the standard form y′′+P(x)y′+Q(x)y=0.
- Substitute the series solution and its first two derivatives into the DE in 1. That is, substitute the terms y=∞∑n=0cnxny′=∞∑n=1ncnxn−1y′′=∞∑n=2(n−1)cnxn−2 into the equation to get: ∞∑n=0(n−1)cnxn−2+P(x)∞∑n=0ncnxn−1+Q(x)∞∑n=0cnxn=0
- Distribute the factors P(x) and Q(x) so they are inside the sigma, then combine like terms, in particular the x terms.
- Get the sigmas "in phase" by making an appropriate substitution to cause the exponents of x in each series to be equal.
- Make the starting index of each series equal by, for instance, evaluating the first few terms of those series with smaller starting indices, or, say, by lowering the index of a series if it does not change the series.
- Combine the sigmas into one and factor out the x term.
- Use the identity property on p. 269 to derive the recurrence relation and other equations that give the coefficients of x in terms of 0.
- Find as many of the first terms of the series as desired, and use the recurrence relation and other equations in 7 to express the coefficients of each term in terms of either c0 or c1.
- Factor out the coefficients c0 and c1 so the series are written in the form y=c0y1(x)+c1y2(x) This is the general solution of the DE in 1.
Step 2 is justified by theorem 6.1 which says that the second-order linear DE y′′+P(x)y′+Q(x)y=0 has at least two linearly independent solutions of the form y=∞∑n=0cnxn.
Regular and Irregular Singular Points of DE’s with Polynomial Coefficients
Singular Point
If a2(x), a1(x), and a0(x)
are polynomials with no common factors, then the point x0
is a
Procedure for determining whether a singular point of a linear, second-order DE with polynomial coefficients is regular or irregular.
- Put the DE a2(x)y′′+a1(x)y′+a0(x)y=0 into the standard form y′′+a1(x)a2(x)y′+a0(x)a2(x)y=0 where a0, a1, and a2 are polynomials.
- Completely factor (under the set of complex numbers) the polynomials a0, a1, and a2 into n distinct linear factors of the form x−x0 (where n is the degree of the polynomial being factored) and reduce the rational functions a1(x)a2(x) and a0(x)a2(x) to lowest terms. (Such a factorization is guaranteed by the fundamental theorem of algebra.)
- For each factor x−x0 of a2, x0 is a regular singular point if and only if the factor occurs at most once in the reduced form a1(x)a2(x)y′ and at most twice in the reduced form of a0(x)a2(x)y †.
- x0 is an irregular singular point if and only if it is a singular point but is not regular.
† According to Dr. Soash, the factor x−x0 of a2(x) need not appear in the reduced form of either of these terms for the condition to be satisfied, i.e. for x0 to be a regular singular point. That is to say, he claims that the definition of singularity also applies when a2(x), a1(x), and a0(x) are polynomials with common factors. The book’s definition of singularity for polynomial coefficients does not comment on this case, however, it only defines it for the case when they have no common factors. See bottom of p. 271.
Series Solutions of Linear Equations About a Singular Point (Method of Froebenius)
Procedure:
- Identify a regular singular point x0.
- Substitute y=∞∑n=0cnxn+r,y′,y′′ into the original DE.
- Rewrite to find indicial equation and recurrence relation.
-
Cases
Case 1: Roots do not differ by a positive integer y1(x)=∞∑n=0c0xn+r,c0≠0y2(x)=∞∑n=0c1xn+r2,c1≠0
Case 2: Roots differ by a positive integer, (|r1−r2|∈N) y1(x)=∞∑n=0cnxn+r,c0≠1y2(x)=cy1(x)lnx+∞∑n=0bnxn+r2,b0≠0
Case 3: Equal indicial roots (r1=r2) y1(x)=∞∑n=0cnxn+r1,c0≠0y2(x)=y1(x)lnx+∞∑n=0bnxn+r1
- Use recurrence relation to determine the cn′s for the ∑′s.