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18.03 Class 3, Feb 13, 2006
First order linear equations: Models
Vocabulary:
Coupling constant, system, signal, system response,
Models: banks, mixing, cooling, growth and decay.
Solution in case the equation is separable;
general story deferred to Class 4.
[1] If I had to name the most important general class of differential
equations it would be "linear equations." They will occupy most of this
course.
Today we look at models giving first order linear equations.
Definition: A "linear ODE" is one that can be put in the "standard
form"
___________________________



x' + p(t)x = q(t)
___________________________

When t = time is the independent variable, the notation
xdot
is often
used. In these notes I'll continue to write
x'
however.
[2] Model 1. Bank account:
I have a bank account. It has
x
dollars
in it.
x
is a function of time. I can add money to the bank and make
withdrawals.
The bank is a system.
It pays me for the money I deposit! This is
called interest. In the old days a bank would pay interest monthly:
Then
Delta t = 1/12
and
x(t + Delta t ) = x(t) + I x(t) Delta t
[ + .
... ]
I has units (year)^{1} . These days I
is typically about 2% = 0.02 .
You don't get 2% each month! you get 1/12 of that.
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 Winter '08
 Staff
 Linear Equations, Equations

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