Page 283 - Schaum's Outline of Differential Equations

266 LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS [CHAP. 27

or

Equating each coefficient to zero, we have
2a 2 +a 0 =0, 6a 3 +^ = 0, 12a 4 +a 2 =0, 20a 5+a 3 = 0, ...
In general

which is equivalent to

This equation is the recurrence formula for this problem.

27.6. Use the power series method to find the general solution near x = 0 of y" + y = 0.
Since this equation has constant coefficients, its solution is obtained easily by either the characteristic equation
method, Laplace transforms, or matrix methods as y = c l cos x + c 2 sin x.
Solving by the power series method, we successively evaluate the recurrence formula found in Problem 27.5
for n = 0, 1, 2,..., obtaining

Recall that for a positive integer n, n factorial, which is denoted by n\, is defined by
«!=«(«-!)(«-2) •••(3)(2)(1)
and 0! is defined as one. Thus, 4! = (4)(3)(2)(1) = 24 and 5! = (5)(4)(3)(2)(1) = 5(4!) = 120. In general,
n\ =n(n- 1)!.
Now substituting the above values for a 2, a 3, a 4 ,--- into Eq. (27.5) we have

But