68 BIRKHOFF AND LANGER. 



The matrix of functions G(x, t) is known as the Gree?i's function for 

 the homogeneous system (6). By a precisely similar method the 

 Green's function H(x, t) for the related system (7) may be derived, the 

 general solution of the non-homogeneous system 



f -Z'(x) = - -Z(x)A(x) + -D(x) 

 { ' \ -Zip) Wa- 1 + -Z(b) W h - l =0 



being given in terms of H(x, t) by the formula 



b 



(15) -Z{x) = I ■ D(x) H(x, t) dt. 



a 



Theorem: If G (x, t) and H(x, t) are the Green's functions for sys- 

 tems (6) and (7) respectively, then 



G(x, t) + H(t, x) = 0, t^x. 



Proof: 6 Let B- and D be any two vectors of the types indicated 

 and consider the two systems 



r- = AY- +B-, W a Y{a)-+W b Y(b)-=0, 



■Z' = --ZA+-D, -Z(a) W a ~ l + -Z(b) W b ~'= 0. 



Multiplying the differential equations respectively by Z on the left 

 and Y • on the right and adding we obtain 



■ZY-' + Z'Y- = ZB- + DY-, 



an equality which upon integration yields 



(16) 







ZY- \ b a = J -Z{t) B(t)-dt + J -D(x) Y(x)-dx. 



Now from the boundary conditions we have 



•Z(b) = - -Z{a) W a ~ l W b , Y(b)- = - JIV 1 W a Y(a)-, 



whence 



■Z(b) Y(b)- = -Z{a) Y{a)-, 

 i.e. ZY- b = 0. 



a 

 6 The proof by direct computation is not difficult though somewhat laborious. 



