BOUNDARY PROBLEMS AND DEVELOPMENTS. 07 



Defining G(x, t) by the relations 



f \ Y h (x) Z h (t) when /, < x 

 G(x, t) = < 



[ — \ Y h (x) Z h (t) when t > x, 



we may write 



Y(x)> = / G(x,t) B(t)- dt. 



a 



In accordance with formula (5), therefore, the general solution of 

 the differential equation (10) is given by the equation 



b 



Y(x)- = I' G(x, t) B (0- dt + Y h (x)C- . 



a 



The substitution of this form in the boundary conditions yields the 

 equation 



J [W a Q(a,t) + W b GQ>,t)} B(t)-dt+{W a Y h (a) + W b Y h {b)}C- = 



a 



for the constant vector C- . Multiplying by the inverse of the matrix 



A= \W a Y h (a) + W h Y h (b)\, 

 (A -1 exists since system (6) is incompatible) we see that 



6 



C- = - A- 1 



J \W a G{a,i) + W b G (b, 0} B(t)>dt. 



It follows that the general solution of system (10) is, in terms of a 

 matrix G(x, t) which is defined by the formula 



(12) G(x, t) = G(x, t) - Y h (x) A- 1 { W a G(a, t) + W \ Q{b, t) ) , 

 given by the expression 



6 



(13) Y(x) = J G(x,t)B(t)- dt. 



