18.03 Class 3
, Feb 8, 2010
First order linear equations; systems and signals perspective
[1] First order linear ODEs
[2] Bank Accounts; rate and cumulative total
[3] Systems and signals language
[4] RC circuits
[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 I'll show you how to model two real world systems by first order linear
equations. Both of them involve systems evolving in time. The independent
variable is time,
t . If we write
x
or
x(t)
for the dependent variable,
we'll write
xdot
for its timederivative. In these ascii notes I'll
continue to write
x'
the deriviative, though.
Definition: A "linear first order ODE" is one that can be put in the
"standard form"
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r(t) x'(t) + p(t) x(t) = q(t)

_____________________________________
There is a general analytic method for solving this equation. I'll talk
on Wednesday about that. For now note that if
p, q, and r
are *constant*
then the equation is separable 
r dx / ( q  px ) = dt
Example 0: From recitation: Oryxes with constant growth rate, hunting allowed
x' = kx  h
h = harvest rate
as in Recitation 1. Notice that in the physical system both
k
and
h
can vary with time, and that the modeling process is fine with that.
The slide shows an Oryx.
[2] Bank account:
I have a bank account. It has
x
dollars in it.
x
is a function of time. I can deposit money in the account and make
withdrawals from it. The bank pays me rent for the money I deposit!
This is called interest.
In the old days a bank would pay interest at the end of the month on
the balance at the beginning of the month. We can model this
mathematically:
With
Delta t = 1/12 , the statement at the end of the month will read:
x( t + Delta t ) = x(t) + I x(t) Delta t
+ [deposits  withdrawals between t and t+Delta t]
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View Full DocumentI has units (year)^{1} . These days
I
is typically very small, say
1% = 0.01 .
You don't get 1% each month! you get 1/12 of that.
You can think of a withdrawal as a negative deposit, so I will call everything
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 Spring '09
 vogan
 Linear Equations, Equations

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