1 68 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



where X ifJL is the repeated root and 



C k = 



tan 6 k = ~ 



The particular integral. 



\ r) 2 



8.110 The operator D stands for , D 2 for v ........ 



The differential equation 8.100 may.be written: 



(D n + diD"- 1 + 02 Z)"- 2 + ..... + a n )y = f(D)y = V(x) 



f(D) = (D- \,)(D - X 2 ) ...... (D- X B ), 



where Xi, X 2 , ...... , X n are determined as in 8.101. The particular integral is: 



y = e^ lx I e ( ^-^ x dx I (**"**) *<& ....... \ e~^n(^V(x)dx. 



8.111 rryrr may be resolved into partial fractions: 



i N, N, N n 



f(D) D-*TD-\2 D-\ n 



The particular integral is: 



= Nie^ ix \e-^ x V(x}dx + N 2 e^ x fe~^ x V(x}dx + ..... 



+ N n e\ x I e- 



THE PARTICULAR INTEGRAL IN SPECIAL CASES 



8.120 V(x) = const. = c, 



c 



y = . 



a n 



8.121 V(x) is a rational integral function of x of the mth degree. Expand 



'r/T^\ m ascending powers of Z>, ending with D m . Apply the operators D, D 2 , 



, D m to each term of V(x) separately and the particular integral will be 



the sum of the results of these operations. 



