Annals of Mathematics

Note on Periodic Solutions of Linear Differential EquationsAuthor(s): Frank UnderwoodSource: Annals of Mathematics, Second Series, Vol. 31, No. 4 (Oct., 1930), pp. 655-656Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1968160 .

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NOTE ON PERIODIC SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS.*

BY FRANK UNDERWOOD.

1. This short note is suggested by, and may, perhaps, be treated as an appendix to, a paper of the above title by Fite,t so that his notation is retained.

If the coefficients in the equation (1) Po Y(71)++p y(n-1) . ...... +Pn-l Y'+ PnY = 0

have the period I and Yi, Y2, *.**, yn form a fundamental system of solutions, yi (x + 1) is also a solution. Hence

(2) y (x +) ail Yj (x) + ai2 y2 (X) + ?+ ai yn (X), i 1, 2, 3, *., n.

The fundamental equation of this substitution is

all - 0) a12 ...... ain a21 a22- ...... a2n

(3) =0.

ani an2 ... ann

Fite considers also the equation

(4) Po Y(n)+ Pi y(n) . . . +pn- Y'+PnY =

where f also has the period 1, but considers only for both (1) and (4) solutions of period 1, though he mentions that if a root of (3) is a kth root of unity, there is a solution of (1) of period ki. It is intended in this note to consider conditions for the existence of solutions of period ki, where k is an integer, and to show that Fite's own methods may be used for this purpose without any serious changes.

2. Owing to the periodicity of the coefficients in (1), yi (x + ki) is also a solution. Hence

iy (x + kl ) AiY (X) + A? 2 y2 (X) + + Ain yn (X)Y ()i 1, 2, 3, ..., n, * Received November 1, 1929. t Fite, Annals of Mathematics, (2), vol. 28 (1926), pp. 59-64.

655

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656 F. UNDERWOOD.

and 8ij is a function of the coefficients ac, in (2). The fundamental equation of this substitution is

811 - @ 812 di 88 Pn 4621 4622- 862n

(6) ..=0.

Aln1 2n ...... Finn

Now if (3) has a root which is a kth root of unity, (6) has a root equal to 1.

The function f, the right hand member of (4), may now be subjected only tQ the condition that it is of period p 1, where k/p is an integer. When this change has been made Fite's work may be repeated and his first theorem restated as:

THEOREM I. If (1) has no solution of period ki, (4) has one, while if (1) has such a solution, (4) may or may not have one.

Similarly Theorems II and III may be restated, with the words "of P+1

period 1" altered to "of period ki" and ..... dx changed everywhere

to ... dx. It is easy to construct simple examples in which solutions of period ki

exist, though those of period I do not. Thus one similar to those given by Fite* is

y"+9y - 8cosx.

Every solution of the corresponding self-adjoint homogeneous equation has the period 2 nJ3, but the non-homogeneous equation has a solution y = cosx, of period 2 a. Thus I -2 /3, k=-3 and all the solutions of the non-homogeneous equation are of period ki, but none of period 1. With the notation used by Fite,t

{'+1 {+2{+ 2nr

J zfdx- S 8 cos3 xcosxdx or J 8sin3 x cosxdx,

neither of which is zero; but

XJi zf dx f+ zf d1 is zero in each case.

* Ibid., p. 62. t Ibid., p. 62, line 1.

UNIVERSITY COLLEGE, NOTTINGHAM.

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