1. Chapter 5 Series Solutions of Linear Differential Equations

2. Outline 5.1 Solutions about Ordinary Points 5.1.1 Review of Power Series 5.1.2 Power Series Solutions 5.2 Solutions about Singular Points 5.3 Special Functions 5.3.1 Bessel Functions 5.3.2 Legendre Functions

3. Solutions about Ordinary Points A power series in x – a is an infinite series of the form – This power series is centered at a – The following series is centered at

4. Solutions about Ordinary Points (cont’d.) Power series characteristics – A power series is convergent if its sequence of partial sums converges (may be checked by the ratio test) – Every power series has an interval of convergence, the set of all real numbers for which the series converges

5. Solutions about Ordinary Points (cont’d.) Power series characteristics (cont’d.) – Every power series has a radius of convergence, R, the set of all real numbers for which the series converges

6. Solutions about Ordinary Points (cont’d.) Power series characteristics (cont’d.) – A function f is analytic at a point a if it can be represented by a power series in x – a with a positive radius of convergence – Power series can be combined through the operations of addition, multiplication, and division

7. Solutions about Ordinary Points (cont’d.) In order to add two series, summation indices must start with the same number and the power of x in each must be “in phase” As written, the first series starts with x 0 while the second starts with x 1 Pulling out the first term of the first series makes both terms start with x 1 Letting k = n – 2 in the first series and k = n + 1 in the second gives matching summation indices Now the series can be added term by term

8. Solutions about Ordinary Points (cont’d.) Consider the linear second-order DE – Divide by to put into standard form – Point x 0 is an ordinary point of the DE if both and are analytic at x 0 – A point that is not an ordinary point is a singular point of the equation

9. Solutions about Ordinary Points (cont’d.) If is an ordinary point of the DE, we can always find two linearly independent solutions in the form of a power series centered at x 0 A series solution converges at least on some interval defined by where R is the distance from x 0 to the closest singular point – This R is the lower bound for radius of convergence

10. Solutions about Singular Points Consider the linear second-order DE – Divide by to put into standard form – Point x 0 is a regular singular point of the DE if both and are analytic at x 0 – A singular point that is not regular is an irregular singular point of the equation

11. Solutions about Singular Points (cont’d.) To solve a DE about a regular singular point, we employ Frobenius’ Theorem – If is a regular singular point of the standard DE, there exists at least one nonzero solution of the form where r is a constant, and the series converges at least on some interval

12. Solutions about Singular Points (cont’d.) After substituting into a DE and simplifying, the indicial equation is a quadratic equation in r that results from equating the total coefficient of the lowest power of x to zero The indicial roots are the solutions to the quadratic equation The roots are then substituted into a recurrence relation

13. Solutions about Singular Points (cont’d.) Suppose that is a regular singular point of a DE and indicial roots are r 1 and r 2 – Case 1: r 1 and r 2 are distinct and do not differ by an integer, then there are two linearly independent solutions

14. Solutions about Singular Points (cont’d.) Suppose that is a regular singular point of a DE and indicial roots are r 1 and r 2 – Case 2: r 1 – r 2 = N where N is a positive integer, then there are two linearly independent solutions – Case 3: r 1 = r 2, then there are two linearly independent solutions

15. Special Functions The following DEs occur frequently in advanced mathematics, physics, and engineering – Bessel’s equation of order v, solutions are Bessel functions – Legendre’s equation of order n, solutions are Legendre functions

16. Special Functions (cont’d.) There are various formulations of the solutions to Bessel’s equation – Bessel’s functions of the first kind – Bessel’s functions of the second kind – Bessel’s functions of half-integral order – Spherical Bessel functions of the first and second kind

17. Special Functions (cont’d.) Legendre polynomials,, are specific nth- degree polynomial solutions The first few Legendre polynomials are