Series Solutions of SOLDEs with Regular Singular Points ECE 6382 Notes are from D. R. Wilton, Dept. of ECE David R. Jackson 1

Frobenius’ Method 2 Our goal is to solve this type of differential equation (SOLDE) using the Frobenius method (series solution). or Ferdinand Georg Frobenius

Frobenius’ Method 3 Un-normalized Normalized (Difficult and not considered further here.)

Frobenius’ Method 4 Ordinary point:  Both solutions are in the form of Taylor series, and correspond to analytic functions.  This is the easiest case, but unfortunately, it is not so common in practice. Taylor series

Frobenius’ Method 5 Regular singular point: Indicial Equation (quadratic equation for  : comes from term with smallest exponent) “Frobenius series” Substitute into DE (  is not usually an integer, a 0  0)

Frobenius’ Method 6 Regular singular point  Both solutions are in the form of a Frobenius series. “Frobenius series”

Frobenius’ Method 7 Regular singular point  The first solution is in the form of a Frobenius series.  The second solution has a Frobenius series added to a term involving the first solution and a ln function. “Frobenius series”

Frobenius’ Method 8 Regular singular point  The first solution is in the form of a Frobenius series.  The second solution is a Frobenius series or has a Frobenius series added to a term involving the first solution and a ln function (either case is possible). “Frobenius series” or Case 3a Case 3b

Frobenius’ Method 9 Helpful formula that is useful for finding the second solution y 2 (x) (This can be derived after some algebra.)

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 10 Choose a = 0 as the expansion point.

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 11

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 12 We then have:

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 13 This is the Bessel equation of order zero. Note: The general Bessel equation of order n is Choose a = 0 as the expansion point.

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 14

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 15

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 16

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 17 Set a 0 = 1: The Bessel function of the first kind, order zero, is defined as:

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 18 Another form:

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 19

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 20 If b 0  0, this last term would generate y 1.

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 21 Note: N 0 (x) is often denoted as Y 0 (x.) Note: The N 0 function has a branch cut on the negative real axis.

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 22 This is the Bessel equation of order 1/2. (This is important in the calculation of the spherical Bessel functions.) Note: The general Bessel equation of order n is Choose a = 0 as the expansion point.

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 23

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 24

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 25 Note: We were successful at generating two solutions using only Frobenius series! Hence

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 26 Hence, we have Note: We don’t need to keep the second term (the sin term) in y 2 (x ), since it is the same as y 1 (x). Also, choose the leading constants to be 1. Bessel functions of 1/2 order: Hence,

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 27 Spherical Bessel functions (of order zero)*: *These are improtant in the solution of the 3D wave equation in spherical coordinates. Hence,

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 28 This is the Bessel equation of order 1. Note: The general spherical Bessel equation of order n is Choose a = 0 as the expansion point.

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 29

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 30

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 31 The Bessel function of the first kind, order one, is defined as: Take a 0 = 1/2:

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 32 We can also do a Frobenius solution to the general Bessel equation of order n (derivation omitted)*: where *In fact, we can even let n  , an arbitrary complex number.

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 33 Another form:

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 34 Note: Trying a Frobenius solution with  2 = -1 will fail. (Try it!)

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 35

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 36 Note: N 1 (x) is often denoted as Y 1 (x.) Note: The N 1 function has a branch cut on the negative real axis.

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 37 where

Series Solutions of Second Order Linear Differential Equations (SOLDEs) 38 where (Schaum’s Outline Mathematical Handbook, Eq. (24.9))