368 PRACTICAL MATHEMATICS 



(b) Let the initial conditions be x = 4 and v = when t = 0. 



at 



But when t = 0, x = 4, and A + B = 4 

 Also, when t = 0, v = 0, and - 2A - 4B = 

 Hence A = 8 and B = - 4 



x = 8e~ 2t - be-* 



= 4,(2e~ 2t - e-*} 



(c) Let the initial conditions be x = 4 and v = 8 when t = 0. 

 x =Ae- 2< + Be- 4 ' 



But when t = 0, a; = 4, and A + B = 4 

 Also, when < = 0, u = 8, and - 2A - 4B = 8 

 Hence A = 12 and B = - 8 



x = I2e~ 2t - 8e-" 



FIG. 124. 



Fig. 124 shows the three relations plotted for values of t betweer 

 and 2. In each case the maximum displacement of the body 

 from the equilibrium position is quickly attained and then the 



