DIFFERENTIAL EQUATIONS 169 



8.122 V(x) = ce kx , 



v __ e kx 



y ~ m 



unless k is a root of f(D) = o. If k is a multiple root of order r of f(D) = o 



cx r e kx 



where 



f(D) =(D- 



8.123 V(x) = c cos (foe + a). 



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



C 



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

 part of 



cx r e i(kx+a) 



fV(ik) ' 



where f (r )(ik) is obtained by taking the rth derivative of f(D) with respect to D, 

 and substituting ik for D. 



8.124 V(x) = csm(kx + a). 



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



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

 part of 



8.125 V(x) 



where X is any function of x. 



y = 



If X is a rational integral function of x this may be evaluated by the method 

 of 8.121. 



8.126 V(x) = c cos (kx + ot)-X, 



where X is any function of x. The particular integral is the real part of 



rp i(kx+a) __ I - V . 



8.127 FO) = c sin (&# + a) -^T. 



The particular integral is the real part of 



_ j re i(kx+a) - 1 - X 



f(D + ik) 



