MISCELLANEOUS APPLICATIONS OF SAMPLED-DATA THEORY 301 
points are plotted on Fig. 11.13, where it is seen that the accuracy is 
improved over that obtained with the 1-sec interval. This is especially 
true in those regions where the second and higher-order derivatives of 
the output are high. 
11.4 Equations with Time-varying Coefficients 
The techniques used in the previous section can be adapted to the case 
of time-varying systems.®!7 These systems are described by differential 
equations containing coefficients which are functions of time. One form 
of equation is 
d7—1 
oe + Gn—1 = + +++ + goy = f(t) (11.36) 
where the various g; are functions of time. If the g; are slowly varying 
c(t) 
1.2 
9 
1.0 »~ 
o= 
0.6 a x T=1/2 sec. 
o T=1 sec. 
—— Exact 

0 1 2 3 4 5 6 7 
Time. sec. 
Fra. 11.13, Exact and approximate response of system in example. 
with time, (11.36) can be expressed in terms of an approximation. In 
general, it is readily shown that the nth derivative d(g,) /dé” is given by 
ga) Son ie ae 


dt” dt® dt»—* 
where the various C;, are the binomial coefficients. If g, is slowly varying, 
the contributions of the derivatives of gn are negligible, so that the 
