366 PRACTICAL MATHEMATICS 



tinuously diminishing amplitude, but in each case the periodic 

 time is the same. 



2TC 



Periodic time = = 2-0944 sees. 

 3 



g 



Frequency = = 0-4775 



2rc 



Case II. When a = b. Let a = b = 2. 

 Then <x= -2 



x = e~ 2t (\ + Et) 



where A and B are constants depending upon the initial con- 

 ditions. 



(a) Let the initial conditions be x = and v = 10 when t = 0. 

 x = e- 2< (A + Et) 



Et) + Be- 2 ' 



H> 



But when t = 0, a; = 0, Then A = 

 Also, when t = 0, v = 10. Then B = 10 

 Hence x = Wte~ 2t 



(b) Let the initial conditions be x = 5 and v = when t = 0. 



x == e~ 2 '(A + B/) 



dx 

 dt 



But when = 0, x = 5, Then A = 5 



Also, when t = 0, v = 0. Then B - 2A = 0, or B = 10 



Hence x = e~ zt (5 + lot) 



= 5e- 2 '(l + 2t) 



(c) Let the initial conditions be x = 5 and v = 10 when t = 0. 



a; = e- 2 '(A + Et) 



v = ^ = _ 2e- 2 '(A + B*) + Be~ 2< 



But when t = 0, x = 5. Then A = 5 



Also, when t = 0, y = 10. Then B - 2A = 10, or B - 20 



Hence x = e~ 2< (5 + 20/) 



