This preview shows pages 1–6. Sign up to view the full content.
Supplemental Lecture 3
Differential Equations
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document III  2
Differential Equations
•
Let
f
:
R
n
→
R
.
•
The equation
is an
n
thorder differential equation
.
=


1
1
)
(
,
,
)
(
),
(
)
(
n
n
n
n
dt
t
x
d
dt
t
dx
t
x
f
dt
t
x
d
III  3
Complete Specification
•
To complete the problem, we need to specify
initial conditions
x
(0),
dx
(0)/
dt
, . . . ,
d
n

1
x
(0)/
dt
.
•
A solution to the differential equation is a
C
n
function
x
:
R
→
R
that satisfies the initial
conditions and such that the derivatives satisfy
the differential equation for all
t
.
•
There will be a unique solution to this
problem.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document III  4
Numerical Approximation
•
We can obtain a numerical approximation to
the solution as follows.
•
Choose a step size
∆
t
> 0.
•
x
(
i
)
(
t
) will be an approximation to
d
i
x
(
t
)/
dt
for
i
= 0, . . . ,
n
.
–
x
(0)
(
t
) approximates the solution
x
(
t
).
III  5
Computing the Approximation
•
For
i
= 0, . . . ,
n
, let
•
For
j
≥
0, assume we have defined
x
(
i
)
(
j
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 01/09/2011 for the course ECON 7230 taught by Professor Feigenbaum during the Spring '10 term at Utah Valley University.
 Spring '10
 Feigenbaum
 Macroeconomics

Click to edit the document details