364 PRACTICAL MATHEMATICS 



When the body is at a distance x feet from its position of 

 equilibrium the resistance of the spring is T lb., while the resistance 



of the medium is kv lb., where k is a constant depending upon 

 the nature of the medium ; both of these forces tend to urge 

 the body back to its equilibrium position. 



/O" \ 



Hence the total resistance to the motion is ( T + kv) lb. 



x , m 



and T + kv = x acceleration 



h g 



d 2 x gk dx g 



or -X2+ ^7 + -r- x = 



at* m at nm 



This can be expressed as 



d 2 x dx , 



-j-s- + 2a -j- + Wx = 



dt 2 dt 



where 2a = and 6 2 = -f- 



m nm 



Let x = Ae * be the solution of this equation. 

 Then a 2 + 2aa + & 2 = 



a+= Va 2 - b* 

 and a, = a + V 2 b z 



The complete solution is x = Ae i< + Be ^' 

 Case I. When 6>a. Let a = 1, and 6 2 = 10. 



Then a = - 1 V^ 



= - 1 Si 

 Then x = AJ- 1 * 3 + 



= e-*(Ae i3t 



= e-'(C cos 3i + D sin 3i) 



where C and D are constants depending upon the initial 

 conditions. 



(a) Let the initial conditions be x = and v = 9 when t = 0. 

 x = e-'(C cos 3t + D sin 3t) 



(if 



v = ^- = - e-*(C cos 3t + D sin 8*) + e~*( - 3C sin 8t + 3D cos 8*) 

 at 



But when f = 0, x = 0. Then C = 

 Also, when t = 0, v = 9. Then D = 3 

 x = 3e~* sin 3/ is the complete solution 



