BOUNDARY PROBLEMS AND DEVELOPMENTS. 59 



Section III. 

 The equation Y'(x) = A(x)Y(x)+ B(x). 

 The Existence Theorem: Given the equation 

 (3) Y' = AY + B, 



where A(x) and B(x) are matrices of continuous functions, a ^ x ^ b, 

 then there exists a unique matrix Y(x) whose elements are functions 

 which are continuous together with their first derivatives, a ^ x ^ b, 

 and which satisfies equation (3) as well as the condition Y(xo) = Yo, 

 the matrix Yo being any prescribed matrix of constants. 

 Proof: By means of the relation 



X 



Y m (x) = F + J U00 F«-i(<) + B(t)} 



dt 



it is possible to define the following infinite sequence of matrices 



Y\{x), Yi{x),. . . Y m (x), . . . which satisfy the relations 



Y[(x) = A(x) Y Q + B(x) 9 Y 1 (x ) = F , 



Y' 2 (x) = A{x) Y,{x) + B(x), Y 2 (x ) = F , 



Y m '{x) = A(x) F m _!(.r) + B(x), Y m (*o) = F . 



Then setting Y m (x) - Y m -i(x) = U m (x) 



in the identical equation 



Y m {x) = F + (Y 1 (x)-Yo)+(Y 2 (x)-Y 1 (x))+ . . .+(Y m (x)-Y m .i(x)) 



we have 



Y m {x) = F + Ui(x) + U 2 (x) +...+ U m {x). 



Moreover, U m (xo) = 0, 



while U m '(x) = Y m '(x) - Y^'ix) 



= A(x) {Y m ^(x) - y m _ 2 (.r)} 

 = A (x) U m -i(x) . 



