April, 1916] Derived Solutions of Differential Equations 233 



(1) d 



dva 



(gmx) 



(17) 2 (m-2) (17) (2) 34 



(b) For (D2 + a2) y = sin (mx) 



(1) _ sin (mx) _ sin (mx) 

 ^ ~ D2+a2 ~ -m"-+a2 



This fails when m = a, but 



(1) d , . , .. 



— — (sm (mx) ) , 



_ dm _ X cos (mx) _ x cos (ax) 



^ ~ 5 X " I 2^ ~ ~2m ~ -2a 



5m 



(2) ^^ _ cos (mx) _ cos (mx) _ x sin (mx) _ x sin (ax) , 

 when m = a. 



^ D2+a2 a--m- 2m 2a 



(3) _ sin mx _ sin mx 



^ ~ (D2+9) (D2+-i) ~ (9-m2) (4-m2) 



For m = 2, 



_ X cos mx _ — X cos 2x 



^ ~ 5(-2m) 20 



For m = 3, 



_ X cos mx _ X cos 3x 



^ ~ (-2m) (-5) ~ 30 



(4) sin mx 



^ "" (D-^+4)2 ' '' '^^ 



sin mx x cos mx — x^ sin mx 



(4-m2)2 (-4m) (4-m2) (-4m) (-2m) 



— x^ sin 2x 

 32 



