DIFFERENTIAL EQUATIONS 17 1 



8.201 The complementary function. 



y = CiX^ + C 2 X^ + ..... + Cn X\ 



where Xi, X 2 , ...... , X n are the n roots of 



F(X) = o 

 if the roots are all distinct. 



If Xfc is a multiple root of order r, the corresponding terms in the comple- 

 mentary function are: 



x* k lt>i + b 2 log * + b 3 (log *) 2 + . . . . + b r (log x) r ~ 1 }. 



If X = v zV is a pair of complex roots, of order r, the corresponding terms 

 in the complementary function are: 



*"{[4i + A 2 log x + As (log *) 2 + .... + A r (log a)'- 1 ] cos (*> log x) 



+ [J5i + 2 log * + .63 (log x) 2 + .... + J5 r (log a;)"- 1 ] sin (v log *) } . 



8.202 The particular integral. 



If 



= ((9-X 1 )((9-X 2 ) ..... (0-\n), 



8.203 The operator ~r may be resolved into partial fractions: 



N 2 N n 



\ n -\ i ..... T 



F(9) ~ d-\^ 0-\ 2 ^ ^ B - \ n ' 



y 



The particular integral in special cases. 

 8.210 V(x) = cx k , 



unless k is a root of F(9) = o. 



If k is a multiple root of order r of F(0) = o. 



c (log *) r 



Ft'KW : 



where F (r )(k) is obtained by taking the rih derivative of F(0) with respect to 6 

 and after differentiation substituting k for 6. 



