18.03 Class 32
, April 26 , 2010
Linear first order systems: Introduction
1. Elimination
2. Matrices
3. Antielimination: companion matrices
4. Shakespeare
[1]
There are two fields in which rabbits are breeding like rabbits,
Farmer Jones's field contains
x(t)
rabbits, Farmer McGregor's
field contains
y(t)
rabbits. Breeding like rabbits in this case
means that Jones's show a net growth rate of .7 rabbits per rabbit per
month, and McGregor's show .6. They can also hop over the hedge between
the fields, and since the grass is greener in Farmer McGregor's field,
Jones's rabbits jump over a the rate of .2 per month, MacGregor's at the
rate of .1 per month.
Flow diagram:
.5
.5
______
______
/
\
/
\
/

.2

\
\

 > 

/
>
Jones


McGregor
<

 < 


.1

So the equations are
x'
=
.5x  .2x + .1y
=
.3x + .1y
(1)
y'
=
.5y  .1y + .2x
=
.2x + .4y
(2)
This is a linear SYSTEM of equations, homogeneous, and it seems to be
impossible to solve, since you need to know
y
to solve for
x
and
you need to know
x
to solve for
y.
We *can* solve, though: use (1) to express
y
in terms of
x :
y = 10 x'  3 x
and then plug this into (2):
10 x"  3 x' = .2 x + 4x'  1.2 x
10 x"  7 x' +
x = 0
or
x"  .7 x' + .1 x
=
0
This is a SECOND ORDER ODE , which we can solve:
s^2  .7s + .1
=
0
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View Full Documenthas roots
r1 = .5
and
r2 = .2 .
(Remember,
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 Spring '09
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 Matrices, Romeo and Juliet, Juliet, Romeo

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