1 70 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



8.128 V(x) = ceP* cos (kx + a). 



If (f$ + ik) is not a root of f(D) = o the particular integral is the real part of 



ce i(kx+a) 



If (/3 + ik) is a multiple root of order r of f(D) = o the particular integral is 

 the real part of 



where / (r > (/3 + i) is formed as in 8.123. 



8.129 V = c$- x sin (kx + a). 



If (j8 + *') is not a root of f(D) = o the particular integral is the real part of 



ik) 



If (j8 + ik) is a multiple root of order r of f(D) = o the particular integral is the 

 real part of 



8.130 V(x) = x m X, 



where X is any function of x. 



The series must be extended to the (w + i)th term. 



8.200 Homogeneous linear equations. General form: 



Denote the operator: 



x = 6 



^, m *" Q ( /) \(& \ //) i \ 



X j == C7IC7 I)(C7 2) ..... (C7 W ~j~ ly. 



The differential equation may be written: 



F(d)-y= V(x). 



The complete solution is the sum of the complementary function, obtained by 

 solving the equation with V(x) = o, and the particular integral. 



