## Friday, July 22, 2022

### 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:    fw / 2 p  =  s-1 / 2 p   =  4/ p  s-1

Period =  T = 2 pw  =  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    =   R/L
b) More specifically,  if E  =  Eo 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    =   R/L:

ie at  =  ò at  E dt  +  C

b) Now, if we let E  =  Eo sin w t

To get the solution for the current i:

i=   Eo  [RL sin w t  -    w L2 cos w t /  (2  + w2 L2 )]  + C -at