Differential Equations

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Differential Equations

• Their Origins

Dillon Fadyn Spring 2000
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In The Beginning

• Newton invented differential equations to
  describe physical laws.

Many of the general laws of nature are best
expressed as differential equations.
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Examples
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Newtons Second Law of Motion
The force acting on a body is equal to
the rate of change of
the momentum of the body.
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Population Growth
The Exponential Model
The rate
at which a population grows
is directly proportional to
the size of the population itself.
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Notice
In the exponential model,
is constant.
the relative growth rate
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Relative Growth Rate

• Absolute Growth Rate

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Population Growth
The Verhulst Model
The relative growth rate of a population
is not a constant,
but is a function
of the size of the population.
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In the Verhulst Model
can assume various forms
leading to different models.
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The Logistic Model
A Version of The Verhulst Model
As the population gets larger,
decreases.
the relative growth rate
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Logistic Model
a
P
a/b
of the population.
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Population Growth
The Predator-Prey Model
A nonlinear System of D.E.s
Youre all experts on this now, yes?
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Compare Relative Growth Rates
Exponential
Logistic
Predator-Prey
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More Examples
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LRC Circuits
L for inductance
C for capacitance
Electromotive force (battery)
R for resistance
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Kirchoffs Law
In words
The sum of the voltage drops across the passive
elements in the circuit equals the applied
voltage.
Passive elements
inductor, resistor, capacitor
Applied voltage
what the battery supplies
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Voltage Drops?
Across the inductor
Across the resistor
Across the capacitor
Here I is current, Q is the charge on the
capacitor.
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Special Notes

• The current is the same at all points in the
  simple circuit we have here.
• The capacitor is the only element with a charge
  associated to it.
• The current is the first derivative of the charge
  on the capacitor.

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The Model
A differential equation that describes the
relationships in the circuit
t
Independent variable
Dependent variable
I
, Q
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Theres A Problem
Two dependent variables are o.k. for a partial
d.e. or for a system.
This model should only have one dependent
variable.
Use
to fix the problem.
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Substituting
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The Model for an LRC Circuit
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Spring-Mass Systems
Imagine a mass m suspended from a spring with a
fixed support.
Suppose the whole system is in a damping medium,
like air, or water, or jello.
Suppose further that there is a driving force,
f(t), making the mass oscillate.
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The Model

• x is the displacement of the mass, measured from
  the resting position
• c is a constant depending on the damping medium
• k is the so-called spring constant (from Hookes
  Law)
• t is the independent, x the dependent variable

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Compare
Dependent variable
Dependent variable
Dependent variable
Constant
LRC Circuit Model
Independent variable
Independent variable
Spring-Mass System Model
Constant
Dependent variable
Dependent variable
Dependent variable
Constant
Constant
Constant
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One Model

• Two entirely different applications

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Final Remarks

• We still havent solved a differential equation,
  but now we know what they might be good for.
• You should be finishing the problems in Section
  1.4 in the text.
• What do they do in the text that we didnt do in
  class?