Download Differential Equations as Mathematical Models

Differential Equations
as
Mathematical Models
Population Dynamics
Animal Population
dP
P
dt
dP
 kP
dt
The number of field mice in a certain
pasture is given by the function 200-10t,
where t is measured in years. Determine a
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
 k  P  (200  10t ) 
dt
Newton’s Law of Cooling
Water Temperature
dT
 T  Tm
dt
dT
 k T  Tm 
dt
Water is heated to the boiling point
temperature of 100ºC. The water is then
removed from the heat and kept in a
room which is at a constant temperature
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  22.5  , T (3)  90
dt
T (9)  ?, T (?)  50.
Spread of Disease
dNi
 Ni Nu
dt
dNi
 kNi Nu
dt
Suppose a student carrying a flu virus
returns to an isolated college campus of
1000 students. Determine a differential
equation governing the number of people
Ni who have contracted the flu if the rate
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 1000  N i 
dt
Chemical Reactions
dx
   x    x 
dt
dx
 k   x    x 
dt
Two chemicals, A and B, react to form
another C. It is found that the rate at
which C is formed varies as the
product of the instantaneous amounts
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 for
dC
 k 10  23x  20  3x  ,
dt
C (0)  0, C (20)  6
Mixtures
dA
 Rin  Rout
dt
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 wellstirred mixture leaves at 4 gal/min. Find the
amount of salt in the tank at any time. is is
dA
4A
 3(1.5) 
dt
10  t
dA
4A
 4.5 
, A(0)  2
dt
10  t
Torricelli’s Law
V  Ah s

dV
dt
A
ds
h dt
 Ahv
dV
  Ah 2 gh
dt
dh
Aw
  Ah 2 gh
dt
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
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
water at time t if the radius of the
cylinder is 2 ft and that of the hole is 2
in. Assume g = 32 ft/sec.
dh
Ah
  Aw 2 gh
dt
dh
  2442  64h   18h
dt
Series Circuits
dI
L  RI  E
dt
dq 1
R  qE
dt C
An inductor of 0.5 henry is connected
in series with a resistor of 6 ohms, a
capacitor of 0.02 farad, a generator
having alternating voltage given by 24
sin(10t), t  0, and a switch. Find the
charge and current at time t if the
charge on the capacitor is zero when
the switch is closed at t=0.
q  12q  100q  48sin(10t )
2
d q
dq 1
L 2 R  qE
dt
dt C
Falling Bodies
dv
m  mg  kv
dt
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 resistance equal to 1.5,
where  is the speed at any instant
during the fall, that she falls vertically
downward, and that the parachute is
already open when the jump takes
place, describe the ensuing
motion.
dv
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
F
d ( vm )
dt
dv
 v dm

m
dt
dt
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
the chain during the time it is sliding over
the edge?
dy
dy
dm
dt
dt
dt
m   y,
mg  v
dm
dt
m
dv
dt
F
  ;v
 
dy 2
dt
 y
.
 
d2y
dt 2
mg   yg    y    yy
2
yg   y   yy
2
Miscellaneous Models--1
Hanging Cable-general
T sin   W ( x), T cos  H
tan  
d2y
dx 2

dy
dx
1 dW
H dx
 WH( x )
Let a cable or rope be hung from two
points, not necessarily at the same level.
Assume that the cable is flexible so that it
curves due to its load (its own weight,
external forces, or a combination of the two).
Determine a DE that governs the shape of
the cable.
Miscellaneous Models--2
Suspension Bridge
d2y
dx 2
d2y
dx 2

1 dW
H dx
 H ,
y (0)  b, y(0)  0
A flexible cable of small (negligible)
weight supports a uniform bridge.
Determine the shape of the cable.
The fact that the bridge is uniform tells
dw
us that
is a constant, say . Then
dx
the DE that models this problem is as
is seen on the left.
Miscellaneous Models--3
dW
ds

dW
dx
ds
  dx
ds
dx
 1
dW
dx
d2y
dx 2
dW dx
dx ds

 
dy
dx
  1


H
1
2
Let a flexible cable having a constant
density, say , hangs between two fixed
points. Determine the shape of the
cable, if we assume the only force acting
on the cable is its own weight.
,
 
dy
dx
 
dy
dx
Hanging Cable
d2y
2
2
dx 2
,
y (0)  b, y(0)  0

1 dW
H dx
Here we first note thatdW 
ds
is the length of the rope.

where s,
Miscellaneous Models --4
Shape of a Reflector
DPR  PFE  FP0;
POG  2 PFE;
tan( ) 
y
x
dy
dx
 y;
2 y
tan
 tan(2 )  12tan

2

1 y2
2

2 xy  y 1   y  


Find the shape of a reflector that
reflects all light rays coming parallel to
a fixed axis to a single point.