EXPANSION PROBLEMS. 393 



readily established by writing the characteristic equation in the real 

 form 



e-p^+2e'^p^s\n(^pTv - ^ j = 0, 

 and observing the changes of sign of the left-hand member. 



II. Differential Equation of Order v with v — 1 Specialized 

 Boundary Conditions. 



The properties of the series studied in the preceding paragraphs are 

 typical of a large class of expansion problems connected with differ- 

 ential equations of order higher than the second. We shall consider 

 now the differential equation ^^ 



(15) ^ + 2>2(.f)^^+... +P.(^)" + P''w = 0, .^3, 



where the coefficients p are real or complex functions of the real vari- 

 able X, continuous with their derivatives of all orders, in an interval 

 which we may still take as the interval ^ .r ^ tt. The v linearly 

 independent boundary conditions associated with this equation are 

 to be subjected to the essential restriction that v — 1 of them involve 

 the values of the solution and its derivatives only at the point 0, while 

 the remaining condition may involve both end-points or the point tt 

 alone. ^^ These conditions can be reduced by linear combination to the 

 form 



1 8 It is understood that any or all of the derivatives of orders from to ^ — 2 

 may occur in the equation; only the {v — l)th is assumed to be absent. As 

 is well known, a more general differential equation can be reduced to this 

 form by a change of variables; cf. Birkhoff II, p. 373, footnote. The differ- 

 ential equation which Liouville considers in the paper II is specialized in a 

 different way. 



19 Of course it amounts to the same thing, if a number of the conditions 

 involve the derivatives at tt in such a way that these can be eliminated from all 

 but one of the conditions by linear combination. A similar remark is applic- 

 able to the cases treated later. The conditions must include at least one 

 which involves the point tt, otherwise they are equivalent to the set 



it(0) = u'ifi) = ... = w(''-i) (0) = 0, 



and no characteristic solutions are possible. In the system treated by Liouville, 

 the j/th condition involves only the point tt, and the coefficients are subject to 

 further restrictions. 



