.(=') 



CI Kit IT Tlir.OKY .l\n ()/7:7v'.(//(>.V.I/. CIlXl'IAS Wl 



however, at once re<lucil)le io ililTeriMiti.il icinatioiis 1>\ tin- ^.lll>slitll- 

 tion I = dQ/dl, whence they become 



Here, as a matter of convenience, we have written l'G* = '^^*- It 

 is often more convenient, at least at the outset, to deal with equations 

 (3) rather than (1). 



The Exponential Solution 



In taking up the mathematical solution of equations (1), we shall 

 start with the exponential solution. This is of fundamental import- 

 ance, both theoretically and practically. It serves as the most 

 direct introduction to the Heaviside Operational Calculus, and in 

 addition furnishes the basis of the steady-state solution, or the theory 

 of alternating currents. 



To derive this solution we set Ei = Fie^' and put all the other 

 forces £«, . . En equal to zero. This latter restriction is a mere matter 

 of convenience, and, in virtue of the linear character of the equations, 

 invoK'es no loss of generality. 



Now, corresponding to Ei = F\e^', let us assume a solution of the 

 form 



/> = V (; = 1,2..«) 



where Jj is a constant. So far this is a pure assumption, and its cor- 

 rectness must be verified by substitution in the differential equations. 



Now if Ij = Jje^', it follows at once that 



jJj = >^Ij = \Jje 

 and 



/,„.{,- Lj/'. 



Now substitute these relations in equations (1) and cancel the com- 

 mon factor e^'. We then get the system of simultaneous equations 



(\Ln + Rn + l/\Cn)Ji + . . + (XL,„ + /?i,+ 1/XC,h)/» = f i, 



(Xi;, + /?ai+l/XC2,)/,-f-. . + {\L2n + R2n+l/\Cin)Jn=0, 



(4) 



(\Lni+RnX+l \Cnl)Jl + . . + (Mnn+Rnn+\ \C„„)J„=0. 



We note that this is a system of simultaneous algebraic equations 

 from which the time factor has disappeared. It is this that makes 



