1193 



r, and c, being arbitrarv consliiiits. 

 Putting 



t = /?/ 

 tliivS takes tlie form 



z zz^ he •* «" (.4,, f'o.s ,j;»/ + /?„ dn ,vn) du. 



o 

 J„ and i?„ being again arbitrary constants. 

 The general integral of 



d^y dy 



dx^ dx 



therefore may be written 



y =: e^' I e ■^ii" {A,^ cos xu -}- B^ sin .vu) die. 







Choosing 



(—1)" njt ^ (-])"+' n.^ 

 A„ = — -=r COS — - B„ = ~ — sin — 



we get the particular integral 



1 of-- z' ^^^^^ 



V = -— - e^~ I e "^ iC cos I .vu d)t 







which for ,c = 0, reduces to 



n; 



2 

 u 



where 



n/ /— 



— K rr (n even) 



/ 



|2 ■ 



ƒ 



e 4 it"du = 



I ^(^—-r-in odd) 



Now we know that 



" nf 



I ~ nf 



U— 1)2 - («even) 



^"(0)=. ^/ (8) 



(n od-1) 



