CIRCUIT THEORY AND OrERATIONAL CALCULUS 717 



vvhtTc a = R/2L and a)'= l/LC. 



Tilt.' nM)ts of tlu- t-qiiation //(/>) =o .in- tin- roots of tlu- f(|ii.itir>ii 



wlu-iue 



/>! = — «+ Va- — co" = — a + ^, 



pj = — « — \/<«- — oi- = — « — /J. 

 Also //'{/)) =2L{p+a), so that 



and 



l/H(o) = l/Lui' = C. 



Inserting tlicse expressions in tiic I'.xpansion TlK-orcni Soluiion 

 (3S). we get 



g-al/ gfl( g-dt 



^ 2sfAa-8 a + B/' 



2/3AV«-/3 a+/3> 



It is now easy to verify tlie fact tliat this solution satisfies tiie difTer- 

 ential equations and tlic boundary condition Q = o and dQ'dt=o at 

 time t = o. 



If a;>nf, ^ is a pure imaginary 



/3 = /o) V 1 — (a/t"))- = i(^' 

 and 



___ e~"'a)' cos a)'/+a sin &)'/ 



In connection with this problem we note two advantages of the 

 expansion solution, as compared with the power series solution: (1) 

 it is much simpler to derive from the operational equation, and (2) 

 its numerical computation is enormously easier. A table of expo- 

 nential and trigometric functions enables us to evaluate Q for any 

 value of / almost at once whereas in the case of the power series solu- 

 tion the labor of computation for large values of / is ver\' great. A 

 third and very important advantage of the expansion solution in this 

 particular problem is that without detailed computation we can 

 deduce by mere inspection the general character of the function 

 and the effect of the circuit parameters on its form: an advantage 

 which never attaches to the power series solution. 



This last property of the particular solution above is extremely 

 important. The ideal form of solution, particularly in technical 



