Page 286 - Schaum's Outline of Differential Equations

CHAP. 27] LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS 269

2
2
Noting that n + n - k - k = (n - k)(n + k+ 1), we obtain the recurrence formula

27.13. Show that whenever n is a positive integer, one solution near x = 0 of Legendre's equation

is a polynomial of degree n. (See Chapter 29.)
The recurrence formula for this equation is given by Eq. (1) in Problem 27.12. Because of the factor n — k, we
find, upon letting k = n, that a n + 2 = 0. It follows at once that 0 = a n + 4= a n + 6= a n + s = .... Thus, if n is odd, all odd
coefficients a k (k > n) are zero; whereas if n is even, all even coefficients a k (k > n) are zero. Therefore, either y\(x)
or y 2(x) in Eq. (27.4) (depending on whether n is even or odd, respectively) will contain only a finite number of
nonzero terms up to and including a term in x"; hence, it is a polynomial of degree n.
Since a 0 and a 1; are arbitrary, it is customary to choose them so that yi(x) or y 2(x), whichever is the polyno-
mial, will satisfy the condition y(\) = 1. The resulting polynomial, denoted by P n(x), is known as the Legendre
polynomial of degree n. The first few of these are

27.14. Find a recurrence formula for the power series solution around x = 0 for the nonhomogeneous differential
2
equation (x + 4)y" + xy = x + 2.
2
2
Dividing the given equation by x + 4, we see that x = 0 is an ordinary point and that (f> (x) = (x + 2)1 (x + 4) is
analytic there. Hence, the power series method is applicable to the entire equation, which, furthermore, we may
leave in the form originally given to simplify the algebra. Substituting Eqs. (27.5) through (27.7) into the given
differential equation, we find that

or

Equating coefficients of like powers of x, we have

In general,

which is equivalent to

(n = 2,3, ...). Note that the recurrence formula (3) is not valid for n = 0 or n = 1, since the coefficients of x° and
1
x on the right side of (1) are not zero. Instead, we use the first two equations in (2) to obtain