ELECTRIC CIRCUIT THEORY 349 



The correct initial and final values of V{t) result for all final values 

 of «<!; so that approximately 



1 



V{t) = 



l+e-^Io{tr 



This equation, while not exact, except for / = and / very large, shows 

 faithfully the general character of V{t) and the way it approaches its 

 final value unity. For large values of t 



6-'/o(/) = l/v'2,r7 

 whence 



Approximations of the foregoing type are not always possible and 

 may not be of sufificient accuracy. I shall therefore give next a 

 method of numerical solution which is generally applicable to integral 

 equations of this type and works quite well in practice. We shall 

 write the integral equation in the more general form 



ii{x) =fix) + I 'u{x-y)k(y)dy (265) 



where f{x) and ^(3;) are known and u(x) unknown. The method 

 depends on the numerical integration of the definite integral. Let 

 us divide the x scale into small intervals d and for convenience write 



u(nd) =Un 



f(nd)=f„ 



k{nd) =k„. 



Now from the integral equation we have at once 



u(o)=Uo=fo, 



u(d)=Ui=fi-{- I u{d — y)k{y)dy. 

 Jo 



Now if d is taken sufficiently small 



r 



Jo 



whence 



and 



u{d-y)k{y)dy = — [mi^o + «o^i], 



Wi =/i+ y biiko-\-iioki\ 



"^=r^i;^t-^^+"°^^'^/2] 



