Docstoc

Differential Equations as Mathematical Models

Document Sample
Differential Equations as Mathematical Models Powered By Docstoc
					Differential Equations
          as
Mathematical Models
    Population Dynamics
           Animal Population
              The number of field mice in a certain
dP
   P
              pasture is given by the function 200-10t,
              where t is measured in years. Determine a
dt            DE governing a population of owls that
              feed on the mice if the rate at which the
              owl population grows is proportional to
              the difference between the number of
              owls and field mice at time t.
dP
    kP      dP
dt               k  P  (200  10t ) 
             dt
     Newton’s Law of Cooling
                     Water Temperature
                    Water is heated to the boiling point
dT                  temperature of 100ºC. The water is then
    T  Tm         removed from the heat and kept in a
                    room which is at a constant temperature
dt                  of 22.5ºC. After 3 minutes the water
                    temperature is 90ºC. Find the water
                    temperature after 9 minutes. When will
                    the water temperature be 50ºC?
dT
    k T  Tm    dT
                        k T  22.5  , T (3)  90
dt                  dt
                   T (9)  ?, T (?)  50.
         Spread of Disease
                  Suppose a student carrying a flu virus
                  returns to an isolated college campus of
dNi               1000 students. Determine a differential
     Ni Nu       equation governing the number of people
                  Ni who have contracted the flu if the rate
 dt               at which the flu spreads is proportional
                  to the number of interactions between
                  the number of students that have the flu,
                  Ni, and those that do not have it yet Nu.

dN i
      kN i N u      dN i
                           kN i 1000  N i 
 dt                   dt
           Chemical Reactions
                           Two chemicals, A and B, react to form
                           another C. It is found that the rate at
dx
      x    x 
                           which C is formed varies as the
                           product of the instantaneous amounts
dt                         of chemicals A and B present. The
                           formation requires 2lb of A for each
                           pound of B. If 10 lb of A and 20 lb of
                           B are present initially, and if 6 lb of C
                           are formed in 20 min, find the amount
                           of chemical C at any time. 2 lb of A
dx                         for
    k   x    x    dC
dt                              k 10  23x  20  3  ,
                                                     x
                           dt
                           C (0)  0, C (20)  6
             Mixtures
                  A tank has 10 gal brine having 2 lb of
                  dissolved salt. Brine with 1.5 lb of salt per
                  gallon enters at 3 gal/min, and the well-
                  stirred mixture leaves at 4 gal/min. Find the
                  amount of salt in the tank at any time. is is
dA
    Rin  Rout
dt                  dA             4A
                        3(1.5) 
                    dt            10  t
                    dA          4A
                        4.5         , A(0)  2
                    dt         10  t
            Torricelli’s Law
  V  Ah s
                     A right-circular cylindrical tank leaks
                     water out of a circular hole at its bottom.
                     If friction and contraction of the water
                     stream near the hole reduce the volume

 dV  Ah ds  Ahv
  dt      dt
                     of the water leaving the tank per second
                     to: cA 2 gh , where c  0.6
                              h

                     Determine a DE for the height, h, of the
 dV                  water at time t if the radius of the
      Ah 2 gh      cylinder is 2 ft and that of the hole is 2
 dt                  in. Assume g = 32 ft/sec.

                       dh
                            Aw 2 gh
                              Ah


   dh                  dt
Aw      Ah 2 gh      dh
   dt                       24 64h   18
                               4
                                 2
                                            h

                       dt
             Series Circuits
                     An inductor of 0.5 henry is connected
 dI
L  RI  E           in series with a resistor of 6 ohms, a
                     capacitor of 0.02 farad, a generator
 dt                  having alternating voltage given by 24
                     sin(10t), t  0, and a switch. Find the
                     charge and current at time t if the
 dq 1                charge on the capacitor is zero when
R  qE              the switch is closed at t=0.
 dt C
                    q  12q  100q  48sin(10t )
  2
 d q  dq 1
L 2 R  qE
 dt   dt C
       Falling Bodies
               A sky diver with a parachute falls
               from rest. Let the combined
               weight of the sky diver and
               parachute be 200 lb. If the
               parachute encounters an air
 dv            resistance equal to 1.5, where 
m  mg  kv    is the speed at any instant during
               the fall, that she falls vertically
 dt            downward, and that the parachute
               is already open when the jump
               takes place, describe the ensuing
                 dv
               motion.
              25     800  6v, v(0)  0
                  dt
              x  0.24 x  32  0, x(0)  0, x(0)  0
     Newton’s 2nd Law of Motion
                                A uniform chain of length L and linear
                                density  lies in a heap on an edge of a
                                smooth table and starts sliding over the
                                edge. Get a DE that governs the motion of
F   d ( vm )
        dt       v dm  m dv
                    dt     dt
                                the chain during the time it is sliding over
                                the edge?

                                 m   y,         dm
                                                  dt   ;v dy
                                                              dt
                                                                          dy
                                                                          dt   .
mg  v          dm
                dt   m   dv
                          dt    F         dy 2
                                              dt      y      
                                                                d2y
                                                                dt 2


                                mg   yg    y    yy
                                                                 2



                                 yg   y   yy
                                                 2
                  Miscellaneous Models--1

                               Hanging Cable-general
                               Let a cable or rope be hung from two
                               points, not necessarily at the same
                               level. Assume that the cable is flexible so
T sin   W ( x), T cos  H   that it curves due to its load (its own weight,
tan        dy
             dx    WHx )
                     (         external forces, or a combination of the two).
                               Determine a DE that governs the shape of
d2y
dx 2
          1 dW
           H dx
                               the cable.
          Miscellaneous Models--2
                       Suspension Bridge
   d2y
   dx 2
             1 dW
              H dx
                       A    flexible   cable of  small
                       (negligible) weight supports a
                       uniform bridge. Determine the
                       shape of the cable.



                       The fact that the bridge is uniform
d2y
dx 2
       ,
        H              tellsdw that
                             us
                            dx
                                         is a constant, say
                       . Then the DE that models this
y (0)  b, y(0)  0   problem is as is seen on the left.
               Miscellaneous Models--3
                                          Hanging Cable
dW
 ds       dW dx
            dx ds   
                                          Let a flexible cable having a constant
dW
 dx      dx
           ds
                                          density, say , hangs between two
                                          fixed points. Determine the shape of
                                          the cable, if we assume the only
                     
                          2
 ds
        1         dy
                              ,           force acting on the cable is its own
 dx                 dx
                                          weight.
                                                     d2y
                                                            1 dW
                      
                                  2
 dW
  dx     1            dy
                         dx
                                                     dx 2    H dx


                                          Here     we       first  
                                                                dW  note      that
                      
                                  2                              ds
           
d2y
dx 2
          H    1       dy
                         dx           ,   where s, is the length of the rope.

y (0)  b, y(0)  0
             Miscellaneous Models --4
                                       Shape of a Reflector
     DPR  PFE  FP0;                  Find the shape of a reflector that
                                       reflects all light rays coming
     POG  2         PFE;              parallel to a fixed axis to a single
                                       point.

tan( )     dy
             dx    y;
y
x    tan(2 )  1tan 2  1 y2
                  2 tan       2y




     2 xy  y 1   y 2 
                           

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:9
posted:3/23/2012
language:English
pages:14