Chapter 13
Ordinary Differential
Equations
Mathematical models in many different fields.
Systems of differential equations form the basis of mathematical models in a
wide range of fields – from engineering and physical sciences to finance and biological
sciences. Differential equations are relations between unknown functions and their
derivatives. Computing numerical solutions to differential equations is one of the
most important tasks in technical computing, and one of the strengths of
Matlab
.
If you have studied calculus, you have learned a kind of mechanical process for
differentiating functions represented by formulas involving powers, trig functions,
and the like. You know that the derivative of
x
3
is 3
x
2
and you may remember that
the derivative of tan
x
is 1 + tan
2
x
. That kind of differentiation is important and
useful, but not our primary focus here. We are interested in situations where the
functions are not known and cannot be represented by simple formulas.
We will
compute numerical approximations to the values of a function at enough points to
print a table or plot a graph.
Imagine you are travelling on a mountain road. Your altitude varies as you
travel. The altitude can be regarded as a function of time, or as a function of longi
tude and latitude, or as a function of the distance you have traveled. Let’s consider
the latter. Let
x
denote the distance traveled and
y
=
y
(
x
) denote the altitude. If
you happen to be carrying an altimeter with you, or you have a deluxe GPS system,
you can collect enough values to plot a graph of altitude versus distance, like the
first plot in figure 13.1.
Suppose you see a sign saying that you are on a 6% uphill grade. For some
Copyright c 2009 Cleve Moler
Matlab
R
is a registered trademark of The MathWorks, Inc.
TM
August 8, 2009
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Chapter 13.
Ordinary Differential Equations
6000
6010
6020
6030
6040
6050
6060
6070
950
1000
1050
1100
1150
altitude
6000
6010
6020
6030
6040
6050
6060
6070
20
10
0
10
20
slope
distance
Figure 13.1.
Altitude along a mountain road, and derivative of that alti
tude. The derivative is zero at the local maxima and minima of the altitude.
value of
x
near the sign, and for
h
= 100, you will have
y
(
x
+
h
)

y
(
x
)
h
=
.
06
The quotient on the left is the
slope
of the road between
x
and
x
+
h
.
Now imagine that you had signs every few meters telling you the grade at
those points. These signs would provide approximate values of the rate of change
of altitude with respect to distance travelled, This is the derivative
dy/dx
.
You
could plot a graph of
dy/dx
, like the second plot in figure 13.1, even though you do
not have closedform formulas for either the altitude or its derivative. This is how
Matlab
solves differential equations. Note that the derivative is positive where the
altitude is increasing, negative where it is decreasing, zero at the local maxima and
minima, and near zero on the flat stretches.
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 Spring '11
 Adams
 Differential Equations, Logic, Numerical Analysis, Equations, Ode, ORDINARY DIFFERENTIAL EQUATIONS

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