66 BIRKHOFF AND LANGEIt. 



where Y(x) is any particular solution and Y h (x) is a matrix solution 

 of the homogeneous equation (1). Hence the general solution of the 

 differential equation in (10) is 



Y(x). = Y h (x)C- +T.r). 



The substitution of Y(x) • in the boundary conditions shows it to be 

 also a solution of the system (10) provided only that C ■ satisfies 



W a Y{a) + W b Y(b) + { W a Y h (a) +W h Y h (h)} C- = 0. 



But this relation can be solved for C-, and uniquely determines C- 

 when and only when | W a Y h (a) + W h Y h (b) | z\z 0, that is when 

 system (6) is incompatible. Q. E. 1). 



Assuming then that system (6) is incompatible it is possible to 

 obtain by the following procedure a solution of the equations (10) 

 which is symmetrical with respect to the ends of the given interval 

 a ^ x ^ b. 



Let Yi(x)- and }' 2 (.r)- be any pair of solutions of the differential 

 equation 



(11) }"(•*•)• =A(x) Y(x)- +lB(z): 



Then clearly their sum Fi(.r)- + Y 2 (x)- = Y{x)> is a solution of equa- 

 tion (10). But by formula (4), applied to equation (11), a particular 

 choice of Y\(x)- and Y->{x)- is seen to be 



X 



Yi{z). = hfY h (x)Z h (t)B(t)'dt 



X 



Yi(x)- = hfY h (x)Z h (t)B(t)-dt, 



b 

 from which it follows that 



F(a)' = hi f Y h (x) Z h (t) B(t)-dt + j Y h (x) Z h (t) B(t)>dt 



la b 



is a particular solution of the differential equation (10). 



