Download Calculus and Differential Equations I: Modeling with Differential Equations - Prof. J. Leg and more Study notes Differential Equations in PDF only on Docsity!
Calculus and Differential Equations I
MATH 250 A
Modeling with differential equations
Modeling with differential equations
Calculus and Differential Equations I
Objects in motion
Newton’s law: for an object moving in one dimension
F
m
γ
m
dv dt
where
F
is the sum of forces applied along the positive
x
direction,
m
is the mass of the object,
x
is the position of its
center of mass, and
v
dx
dt
is its velocity.
If the only force is gravity, then
F
mg
if
x
points upward
in the vertical direction. In this case, we have
dv dt
g
which is solved by direct integration.
Modeling with differential equations
Calculus and Differential Equations I
Objects in motion (continued)
In the presence of gravity and friction, we typically have
F
m g
c v
c
0, if the object is moving slowly.
F
m g
c v
2
c
0, if
v
v
It is possible to have other types of friction forces, especially inthe case of solid friction.
If we restrict ourselves to the above examples, we have
dvdt
g
c m
v
, which is linear in
v
dvdt
g
c m
v
2
, which is separable.
For a spring-mass system, we have
F
k
x
x
0
k
Then,
m
d
2
x
dt
2
k
x
x
0
), which is a second order, linear
equation.
Modeling with differential equations
Calculus and Differential Equations I
Mixture problems
These problems typically involve a fluid, of volume
V
t
), in
which a substance is dissolved. The goal is to find the amount A
t
) or the concentration
C
t
A
t
V
t
) of the substance
in the fluid.The general way of addressing such a problem is to write abalance equation for the amount
A
t
) of the substance in the
fluid,
dA
dt
input rate
output rate
Example (#5 page 207): Take a 200-gallon container filledwith pure water. Add a salt concentration with 3 pounds ofsalt per gallon, at a rate of 4 gallons per minute. At the sametime, drain the container at a rate of 5 gallons per minute.Find the amount of salt in the container as a function of time.
Modeling with differential equations
Calculus and Differential Equations I
Cooling and heating
Newton’s law of cooling and heating says that the rate ofchange of the temperature
T
of an object is a linear function
of the difference between
T
and the ambient temperature
T
0
dT
dt
k
T
T
0
k
This equation can be solved as a linear equation, or as aseparable equation, to find
T
t
T
0
κ
exp(
kt
where
κ
is an arbitrary constant.
As expected,
T
T
0
, as
t
Modeling with differential equations
Calculus and Differential Equations I
Compounding interest
If money in a bank account is compounded continuously at arate of
r
percents per year, then in the absence of deposits or
withdrawals, we have
dM
dt
r
M
where
M
is the account balance and
t
is time measured in
years.The above equation describes the exponential growth of
M
After one year, the amount of money in the account is givenby
M
(1) = exp(
r
M
The annual interest rate is therefore larger than
r
100, since
APY
= exp(
r
Modeling with differential equations
Calculus and Differential Equations I
Population dynamics
If
N
is the population density of a region, then one can write
dN
dt
bN
dN
immigration
emigration
assuming that resources are not limited.In the above equation,
b
is the birth rate, and
d
is the death
rate of the population. The growth rate
r
of the population is
given by
r
b
d
If immigration and emigration are given functions of
t
, then
the above equation is linear in
N
Modeling with differential equations
Calculus and Differential Equations I
Population dynamics (continued)
If a population is growing exponentially at rate
r
0, we can
define its doubling time
T
d
ln(2)
r
Note the analogy with the half-life of a substance decayingexponentially at rate
r
T
1
/
2
ln(2)
r
ln(2)
r
If resources are limited, one can expect that
r
will depend on
N
. With
r
α
β
N
α >
β >
0, and in the absence of
immigration or emigration, we have logistic growth
dN
dt
α
N
β
N
2
Modeling with differential equations
Calculus and Differential Equations I
