CIRCUIT THEORY .-INP OPERATION A1. CALCULUS 707 



("(ins«.'c|ui'iitly thf opera! inn. il formiil.i is 

 1 



<?=r>5 



Lt^' + Rp+l/C 



1 1 , R ., \ 



wluTi- It = , and b = 



Z,/)=l+o//)+ft/A= /- I.C 



'\'W\> can In- t'\|).in(ii'<l !)>• llu' Rinmnial TlicorcMn as 



^'=,.>-;'-(^;)+(;4)'-(^f-)"+ ■(■ 



IVrforniini!; the incliratcci operations and collecting in inverse powers 

 ol p, the first few terms of the expansion are: — 



/'/>'-'' p p- p' p* y p'"^ • ■ ■ f 



where fi=a 



Ci = b — a- 



Ci = 2ab-a'^ 



CA=b--Za'b-\-a* 



Ci = 3ab--4a'b+a'' 



Ct = b^ — (]a-b- + oa*b — a* 



We infer therefore that in accordance with the rule of replacing 

 !//>" by /"/n! the solution is: — 



^ 1 j /^ t' t* t' I' I 



^=Z ' 2!~''3!~'T!"^'^T!+'^T! " ■ • f • 



Owing to the complicated character of the coefficients in the expan- 

 sion, the series cannot be recognized and summed by inspection. If, 

 however, we put R = o) then a=o, and the series becomes 



'2!VlC-^ 4!VvZX-/ UIVIC-/ "^ 

 whence 



Q = C\l-cos(t/VLCJ\. 



We have still to verify this solution by comparison with the explicit 

 solution of the differential equation. This is of the form 



Q = C+k,e^'' + k,e^'-' 



