# Differential Equations as Mathematical Models

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```					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.
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 
            

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