Math 334 Lecture #29
§
7.1,7.2,7.3: Introduction to Systems of First Order ODEs
Example.
(Converting a second order ODE into a system of two first order ODEs.)
An IVP for a massspring system is
mu
+
γu
+
ku
=
F
(
t
)
u
(0) =
u
0
,
u
(0) =
u
0
.
Define two new dependent variables
x
1
=
u
and
x
2
=
u .
The derivatives of these new dependent variables with respect to the independent variable
t
are
x
1
=
u
=
x
2
,
x
2
=
u
=
F
(
t
)

γu

ku
m
=
F
(
t
)

γx
2

kx
1
m
.
[The original second order ODE in
u
was solved for
u
and substituted into the second
first order ODE.]
In matrix form, this system of first order ODEs is
x
1
x
2
=
0
1

k/m

γ/m
x
1
x
2
+
0
F
(
t
)
/m
.
This conversion of a second order ODE into a system of first order ODEs illustrates how
to convert of an
n
th
order ODE into a system of
n
first order ODEs.
Example.
(Obtaining systems of first order ODEs through modeling.)
A dissolved chemical is mixed in a coupled system of two tanks, each of which contains
100 gallons of water.
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 Spring '11
 Smith
 Math, Gallon, IVP, Rate equation, Tier One, Order ODE, 4 gal

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