234 The Ohio Journal of Science [Vol. XVI, No. 6, 



(5) sin mx , , -. 



V = wnpn m = 'J. 



- fD^+ '"^^ 



sin mx x cos mx — x'sin mx 



(4-m2)^ 3(-2m) (4-m2)2 |3j-2m)2(4-m2) 

 — x^cosmx x^cos 2x 



3 



|3_(-2m)^ !3_(4) 

 (6) sin mx , , 



x"sin (n— +mx) x"pm (n-+ax) 



Similar treatment when the second member is the cosine. 



Case IV. 



Linear differential equations, constant coefficients, second 



member a constant. 



This is a particular case of (a) Case III. 



d^v dv 



^_5^y +Gv = 4 = 4e- 



dx- dx 



_ 4e''^ _ ^ _ 2, 



■'• ^' ^ D^-TdD+G ~ 6 ~ 3 



when D = 0, is the particular solution. 



Case V. 



Failing case of Case IV. 



d^y - d y 

 dx^ ' dx 



4gmx 



\^ = 



m^ — 5 m 



is the solution when m = 0. 



4x6'"'' 



.-. As in Case III, y = ^; -, when m = 



2m — 



4x 



or, V = —' 



o 



is the particular solution. 



