62 BIRKHOFF AND LANGER. 



However, since a and n are fixed numbers and 8 can be chosen so that 



5< - this leads to a contradiction unless d — 0. It follows that 

 not 



there always exists a finite interval throughout which d = and hence 



throughout which D(x) = 0. Inasmuch as this implies that D(x) = 0, 



a ^ x ^ b, the solutions Y(x) and Y(x) must be identically the same. 



Q.E.D. 



The solution of equation (3) can be easily expressed in terms of the 



solutions Y h (x) and Z^(.r) of the homogeneous equations (1) and (2) 



respectively. Thus, multiplying both sides of equation (3) on the 



left by Zh{x) we have 



dY 

 Z h — = Z h A Y + Z h B, 



dx 



which, in view of equation (2), can be written 



Z h —- = — — — Y + Z n B, 

 dx dx 



i.e. 



7- Z h Y = Z h B. 

 dx 



Integrating we obtain 



X 



Z h Y = C + f Z h (t) B(t) dt, 



a 



while the multiplication of this equation on the left by Y h {x), the 

 solution associated with Z h (x), yields 



X 



(4) Y(x)= Y h (x)C + j Y k (x) Z h (t) B(t) dt. 



a 



This is the general solution of equation (3). In consequence any 

 particular solution Y(x) may be written 



X 



Y(x) = Y h (x) C + / Y h (x) Z h (t) B(t) dt. 



