Ch 5.4: Regular Singular PointsRecall that the point x0 is an ordinary point of the equation

if p(x) = Q(x)/P(x) and q(x)= R(x)/P(x) are analytic at at x0. Otherwise x0 is a singular point. Thus, if P, Q and R are polynomials having no common factors, then the singular points of the differential equation are the points for which P(x) = 0.

0)()()( 2

2

yxRdxdyxQ

dxydxP

Example 1: Bessel and Legendre EquationsBessel Equation of order :

The point x = 0 is a singular point, since P(x) = x2 is zero there. All other points are ordinary points.

Legendre Equation:

The points x = 1 are singular points, since P(x) = 1- x2 is zero there. All other points are ordinary points.

0222 yxyxyx

0121 2 yyxyx

Solution Behavior and Singular PointsIf we attempt to use the methods of the preceding two sections to solve the differential equation in a neighborhood of a singular point x0, we will find that these methods fail.This is because the solution may not be analytic at x0, and hence will not have a Taylor series expansion about x0. Instead, we must use a more general series expansion.A differential equation may only have a few singular points, but solution behavior near these singular points is important.For example, solutions often become unbounded or experience rapid changes in magnitude near a singular point.Also, geometric singularities in a physical problem, such as corners or sharp edges, may lead to singular points in the corresponding differential equation.

Numerical Methods and Singular PointsAs an alternative to analytical methods, we could consider using numerical methods, which are discussed in Chapter 8.However, numerical methods are not well suited for the study of solutions near singular points. When a numerical method is used, it helps to combine with it the analytical methods of this chapter in order to examine the behavior of solutions near singular points.

Solution Behavior Near Singular PointsThus without more information about Q/P and R/P in the neighborhood of a singular point x0, it may be impossible to describe solution behavior near x0.

It may be that there are two linearly independent solutions that remain bounded as x x0; or there may be only one, with the other becoming unbounded as x x0; or they may both become unbounded as x x0.

If a solution becomes unbounded, then we may want to know if y in the same manner as (x - x0)-1 or |x - x0|-½, or in some other manner.

Example 1Consider the following equation

which has a singular point at x = 0. It can be shown by direct substitution that the following functions are linearly independent solutions, for x 0:

Thus, in any interval not containing the origin, the general solution is y(x) = c1x2 + c2 x -1. Note that y = c1 x2 is bounded and analytic at the origin, even though Theorem 5.3.1 is not applicable. However, y = c2 x -1 does not have a Taylor series expansion about x = 0, and the methods of Section 5.2 would fail here.

12

21 )(,)( xxyxxy

,022 yyx

Example 2Consider the following equation

which has a singular point at x = 0. It can be shown the two functions below are linearly independent solutions and are analytic at x = 0:

Hence the general solution is

If arbitrary initial conditions were specified at x = 0, then it would be impossible to determine both c1 and c2.

0222 yyxyx

221 )(,)( xxyxxy

221)( xcxcxy

Example 3Consider the following equation

which has a singular point at x = 0. It can be shown that the following functions are linearly independent solutions, neither of which are analytic at x = 0:

Thus, in any interval not containing the origin, the general solution is y(x) = c1x -1 + c2 x -3.

It follows that every solution is unbounded near the origin.

32

11 )(,)( xxyxxy

,0352 yxyyx

Classifying Singular PointsOur goal is to extend the method already developed for solving

near an ordinary point so that it applies to the neighborhood of a singular point x0.

To do so, we restrict ourselves to cases in which singularities in Q/P and R/P at x0 are not too severe, that is, to what might be called “weak singularities.”It turns out that the appropriate conditions to distinguish weak singularities are

0)()()( yxRyxQyxP

finite. is )()(lim finite is

)()(lim 2

0000 xP

xRxxandxPxQxx

xxxx

Regular Singular PointsConsider the differential equation

If P and Q are polynomials, then a regular singular point x0 is singular point for which

For more general functions than polynomials, x0 is a regular singular point if it is a singular point with

Any other singular point x0 is an irregular singular point.

finite. is )()(lim finite is

)()(lim 2

0000 xP

xRxxandxPxQxx

xxxx

.at analytic are )()(

)()(

02

00 xxxPxRxxand

xPxQxx

0)()()( yxRyxQyxP

Example 4: Bessel EquationConsider the Bessel equation of order

The point x = 0 is a regular singular point, since both of the following limits are finite:

0222 yxyxyx

22

222

0

20

200

lim )()(lim

,1lim)()(lim

0

0

xxx

xPxRxx

xxx

xPxQxx

xxx

xxx

Example 5: Legendre EquationConsider the Legendre equation

The point x = 1 is a regular singular point, since both of the following limits are finite:

Similarly, it can be shown that x = -1 is a regular singular point.

0121 2 yyxyx

0111lim

111lim

)()(lim

,11

2lim1

21lim)()(lim

122

1

20

1210

0

0

xx

xx

xPxRxx

xx

xxx

xPxQxx

xxxx

xxxx

Example 6Consider the equation

The point x = 0 is a regular singular point:

The point x = 2, however, is an irregular singular point, since the following limit does not exist:

02322 2 yxyxyxx

022

lim22

2lim )()(lim

,022

3lim22

3lim)()(lim

022

0

20

20200

0

0

xx

xxxx

xPxRxx

xx

xxxx

xPxQxx

xxxx

xxxx

22

3lim22

32lim)()(lim

22200

xxx

xxxx

xPxQxx

xxxx

Example 7: Nonpolynomial Coefficients (1 of 2)

Consider the equation

Note that x = /2 is the only singular point. We will demonstrate that x = /2 is a regular singular point by showing that the following functions are analytic at /2:

0sincos2/ 2 yxyxyx

xx

xxx

xx

xx sin2/

sin2/,2/

cos2/

cos2/ 22

2

Example 7: Regular Singular Point (2 of 2)

Using methods of calculus, we can show that the Taylor series of cos x about /2 is

Thus

which converges for all x, and hence is analytic at /2. Similarly, sin x analytic at /2, with Taylor series

Thus /2 is a regular singular point of the differential equation.

0

121

2/!)12(

)1(cosn

nn

xn

x

,2/!)12(

)1(12/

cos1

21

n

nn

xnx

x

0

22/!)2(

)1(sinn

nn

xn

x