BOUNDARY PROBLEMS AND DEVELOPMENTS. 



63 



Subtracting this from equation (4), however, we obtain 



Y(x) = f(x) + Y h (x) { C - C), 



from which it is seen that if Y(x) is any particular solution of equation 

 (3) the general solution is given by 



(5) 



Y(x) = Y(x) + Y h (x) C. 



If in particular C is taken as C = in formula (4), the solution which 

 is characterized by the fact that Y(a) — is obtained. This is called 

 the principal solution at x = a. Since the choice of a as a limit of 

 integration is unrestricted, the expression for the principal solution at 

 any chosen point is at hand. 



Section IV. 



The differential system 



Y'(x)- = A(x)Y(x).+ B(x)* 

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



A matrix in which any row (or column) is precisely like every other 

 row (or column) is called a vector. That a particular matrix is a vector 

 is indicated by means of a dot suitably placed in relation to the letter 

 designating the matrix in question, the dot preceding in case it is a 

 vector in which the rows are the same and succeeding in case it is a 

 vector of identical columns. 



Thus 



A = {a,) = 



and 



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