Solutions To Differential Equations Applications Problems

 1) A 5 lb. weight hangs on a spring whose spring constant is 10. The weight is pulled 6 inches further down and released. Find the equation of motion, its period and frequency. If g = 32 ft/sec/sec give the units for the spring constant.

Soln.

Spring constant k = F/x  =  10 =  (5 lb.)/ (0.5 ft. ) 

With units in feet, lbs. for consistency so  units are:  lbs./ ft.

The form of the equation of motion will then be:

m(d²x/dt ²) +  k x   =   mg   

Or:  m(d²x/dt ²) +  10 x   =   mg   

And since W= mg =  5 lb.  and g = 32 ft/sec/sec

m =   5 lb. / 32 ft/sec/sec

 Þ (d²x/dt ²) +  (k/ m)  x   =   g  

Þ  (d²x/dt ²) +  64 x   =   32 

But note angular frequency:

w =  Ö(k/ m)  =    w =  Ö(10  lb./ft/ 5/32 lb) = 8 s^-1

 So we obtain solution:

x =  1/2 cos w t + 1/2  

Or:   x =  1/2 cos 8 t + 1/2  

(N.B. At t = 0,  x = 6 in = 1/2 ft, then pull down another 6 in. or  1/2 ft)

Frequency:    f = w / 2 p  =  8 s^-1 / 2 p   =  4/ p  s^-1 

Period =  T = 2 p/ w  =  2 p/8 s^-1 =  p/4 s

2) A circuit consists of an inductance L and resistance R  connected in series with effective voltage E.  Find a solution for the current i, 

a) Using the integrating factor exp (ò a dt), if   a  =   R/L

b) More specifically,  if E  =  E_o sin  w t

Soln.

(a) From Kirchoff's law we have for the DE:

L (di/dt) + Ri  =  E

The integrating factor is:

exp (ò a dt), so if   a  =   R/L:

ie ^at  =  ò e ^at  E dt  +  C  

b) Now, if we let E  =  E_o sin w t

To get the solution for the current i:

i=   E_o  [RL sin w t  -    w L² cos w t /  (R ²  + w² L² )]  + C e ^-at