188 The Rev. B. Bronwiu on the Integration of 



Substituting these values in the proposed equation, and opera- 

 ting on the result with D, we find 



zsD m u -\(D)D'"u - p mD m u + knit- k\{D)u=~DO. 



\(D)D m +pniD m + A\(D) =0, 

 which gives 



\(D) = -jm»(D" , +*)- , D to , 



and our equation is reduced to 



OT D m « + krsu - DO, or u = (D m + k) " V" 'DO. 

 Now 



D w +A V D'" + /t/ 

 therefore by the known formula 



we easily find 



= (D m + A-)- /, *- 1 (D m + *yo, 

 and therefore 



k= (D m + ky p - 1 x- l (D m + k) p 0. 

 If p be positive, 



x -i(D m + kfO = 0, and M =(D m + A-p' p+I) 0. 

 If p be changed into — p, we have 

 *D m w +^D m - l u + kxu=0, u= (D m + kf - x x~ ' (D m + k) ""0. 

 We must make 



w» + k=(D+k 1 )(p+kj • • • • (D+-U, 



but I shall not stop to reduce these solutions. It may be well 

 to remark, that we cannot immediately operate with the fonn 



OT=a\D + X(D); 

 but 



er=Dff+X(D)=a?D + l+\(D). 



Therefore by suitably changing the form of X(D) w r e can perforin 

 the operations. 



As another example I shall take 



x*T) m u-p{p- l)D m_2 M + kx 2 u=0, 



where p or p—\ is divisible by m. This is taken from page 197 

 of the volume before referred to. Both this and the preceding 

 example will serve to illustrate the manner in which an equation 



