18.03 Muddy
Card
responses,
February
24,
2006
Thank you all for your frank responses.
I’ll try to answer some of the most common
confusions.
1.
A Frst order ODE is “autonomous” if it has the form
y
˙ =
g
(
y
).
Here
˙
y
=
dy/dt
.
y
is a function of
t
, so it’s true that the right hand side depends on
t
; but only through
the dependence of
y
on
t
.
y
˙ =
ty
is not autonomous;
˙
y
=
y
is.
The
function
g
(
y
)
completely
describes
the
diﬀerential
equation.
In
the
natural
growth/decay case,
g
(
y
) =
k
0
y
.
In the logistic case,
g
(
y
) =
k
0
(1
−
(
y/p
))
y
or, with
harvest,
g
(
y
) =
k
0
(1
−
(
y/p
))
y
−
a
.
If you explore the
Phase
lines
Mathlet you will
see other examples.
2.
There were some questions about how I derived these formulas for
g
(
y
).
Even in
the natural growth case, when
g
(
y
) =
k
0
y
, someone made the good point that the
growth rate
k
0
reﬂects not only the birth rate but also the death rate.
In fact it’s the
birth rate minus the death rate; the two processes act against each other.
3.
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This note was uploaded on 01/31/2011 for the course MAT 17A taught by Professor Staff during the Winter '08 term at UC Davis.
 Winter '08
 Staff

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