Ork — Solutions of Differential Equations by Contour Integrals, 123 



§ 3'2. The solution ohtainecl directly from the eqimtions. 



On writing (22), (23), with f as the independent variable, we may 

 obtain from them 



e\ (i - n 

 e\ {I - f) 



<t>n{D')^+ 0i,(Z»')2/j'/^', 



,p,,(D').v+ <p,,{D')y 



dt\ 



^■=(A) 

 </>..(A) 



0, (42) 



since each member of the first column is zero. 



Transforming the integrals as in § (1"3), the left-hand member becomes 



t 



e\t 



{^11 {D'-) - i>u (A)la^ + {^..(DQ - ^n (A)}y 



D' -X 

 (j..i(Z)0-.^2L(A)ia;+ {i,,,{D') ~ 4,,A\)\y 



])' - A 



e-xt\ 



-At 



f.(A) 



<p2, (A) 



<^n (A), 



^-i(A), 



<t>n (^) 

 02. (A) 



e\{t-na-dt'. 



(43) 



Multiplying by rfA/A(A), and integrating round the contour, the last 

 term contributes nothing, as the determinant in front of the integral is 

 A (A). 



Considering the contribution from the first term, at the upper limit, 

 i.e. t' = t, the integrand is of the same form as the initial value of that 

 in (25), except that the cc, y, in it relate to time t instead of to zero time. 

 And the same argument which proved that the initial value of the contour 

 integral in (25) is STrico proves that the integral arising from the first term 

 at the upper limit is 2Trixt. 



And at the lower limit the first term in (43) gives to the contour integral 

 the value stated in (25) to be a solution. 



In the very special case of m = 0, the above argument does not apply, 

 however ; we may then use a similar argument to that used below, when A 

 is " abnormal," in replacing (48) by (52). 



Thus the solution has been obtained directly. 



§ 4. Systems which do not admit zeros as solutions, and systems which 

 hate an " abnormal " determinant. 



I now suppose that the equations to be solved are of types 



<Pu{D)x + i,,,{D)y = f,(t), (44) 



i>,,{D),: + ^,,{D)y^f,'t), (45) 



where the order of A is (usually) less than ni -i- n ; the argument holds, 

 however, when A is normal. 



