Books
in black and white
Main menu
Share a book About us Home
Books
Biology Business Chemistry Computers Culture Economics Fiction Games Guide History Management Mathematical Medicine Mental Fitnes Physics Psychology Scince Sport Technics
Ads

Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

Boyce W.E. Elementary Differential Equations and Boundary Value Problems - John Wiley & Sons, 2001. - 1310 p.
Download (direct link): elementarydifferentialequations2001.pdf
Previous << 1 .. 139 140 141 142 143 144 < 145 > 146 147 148 149 150 151 .. 609 >> Next

lim (x - x0) Q^ ) is finite (6)
X^X0 P (x )
and
lim (x - x0)2 is finite. (7)
)2 R(x ) X0) P (x)
This means that the singularity in Q/P can be no worse than (x - x0)1 and the singularity in R/P can be no worse than (x - x0)2. Such a point is called a regular singular point of Eq. (1). For more general functions than polynomials, x0 is a regular singular point of Eq. (1) if it is a singular point, and if both9
( ) Q(X) d ( \2 R(X)
(x - x0) and (x - x0) (8)
0
9The functions given in Eq. (8) may not be defined at xq, in which case their values at xq are to be assigned as their limits as x ^ xq.
258
Chapter 5. Series Solutions of Second Order Linear Equations
have convergent Taylor series about x0, that is, if the functions in Eq. (8) are analytic at x = x0. Equations (6) and (7) imply that this will be the case when P, Q, and R are polynomials. Any singular point of Eq. (1) that is not a regular singular point is called an irregular singular point of Eq. (1).
In the following sections we discuss how to solve Eq. (1) in the neighborhood of a regular singular point. A discussion of the solutions of differential equations in the neighborhood of irregular singular points is more complicated and may be found in more advanced books.
In Example 2 we observed that the singular points of the Legendre equation
(1 x2) y" 2xy' + a(a + 1)y = 0
are x = 1. Determine whether these singular points are regular or irregular singular points.
We consider the point x = 1 first and also observe that on dividing by 1 x2 the coefficients of y' and y are 2x/(1 x2) and a(a + 1)/(1 x2), respectively. Thus we calculate
2x (x 1)(2x) 2x
lim (x 1)---------~ = lim-------------------= lim--------= 1
x ^1 1 x x ^1 (1 x )(1 + x) x^1 1 + x
and
2 a(a + 1) (x 1)2a(a + 1)
lim (x 1) --------= lim---------------------------
x ^1 1 x x^1 (1 x )(1 + x)
(x 1)(a)(a + 1) A
= lim--------------------------= 0.
x^ 1 1 + x
Since these limits are finite, the point x = 1 is a regular singular point. It can be shown in a similar manner that x = 1 is also a regular singular point.
EXAMPLE
6
Determine the singular points of the differential equation
2x (x 2)2 y" + 3xy; + (x 2) y = 0 and classify them as regular or irregular.
Dividing the differential equation by 2x (x 2)2, we have
31
y +---------- y +---------y = 0,
2(x 2)2 2x (x 2)
so p(x) = Q(x)/P(x) = 3/2(x 2)2 and q(x) = R(x)/P(x) = 1/2x(x 2). The
singular points are x = 0 and x = 2. Consider x = 0. We have
3
lim xp(x) = lim x-------~ = 0,
x^0 ^ x^0 2(x 2)2
lim x2q (x) = lim x2-------= 0.
2x (x 2)
5.4 Regular Singular Points
259
EXAMPLE
Since these limits are finite, x 0 is a regular singular point. For x 2 we have
3 . 3
lim (x - 2)p(x) lim (x - 2) --------- lim
x^2 x^2 2(x - 2)2 x^2 2(x - 2)
so the limit does not exist; hence x 2 is an irregular singular point.
Determine the singular points of
/ \ 2
7 (x - 2) y" + (cos x)y'+ (sin x)y 0
and classify them as regular or irregular.
The only singular point is x /2. To study it we consider the functions
\ / \ Q (x) cos x
(x - -) p(x> (x - 2)
2' P (x) x - /2
and
\2 / \2 R(x)
sin x.
2/ V 2) P (x)
Starting from the Taylor series for cos x about x /2, we find that
cos x (x - /2)2 (x - /2)4
Previous << 1 .. 139 140 141 142 143 144 < 145 > 146 147 148 149 150 151 .. 609 >> Next