MATH 205, SPRING, 2011
Math 205 is really a combination of two courses, linear algebra and di erential equations. The main
reason for this is so that you don't have to take yet another math course, but there are some sound reasons
for it as well.
Di erential Equations
is the basic tool for mathematical modeling of physical systems. The
idea is to turn basic understanding of the forces at work in a system into a precise description of the resulting
behavior. This is the reason why engineers need to study calculus. On the other hand,
Linear Algebra
can be
thought of as merely a set of formal manipulations and theory. However, it is a central part of the modeling
process we need to do di erential equations. It's practically impossible to understand how you work with
di erential equations without using linear algebra. In the nal analysis, though, both of these subjects will
be very useful for your later course work.
1.
FirstOrder Equations
1.1.
Introduction.
I should start o with an explanation of what a
di erential equation
is. A
di erential
equation
is an equation (gee!) involving some unknown function, usually called
y
, maybe some known
functions of the variable (usually
x
), and derivatives of the function
y
. A simple example might be:
y
00
+ 5
y
0

17
y
=
e
x
.
Many texts tend to write a general equation in a confusing way:
F
(
x, y, y
0
, . . . , y
(
n
)
) = 0
.
They just mean to say that it is an equation (the = ) involving
x
,
y
,
y
0
, and so on, up to the
n
th
derivative
y
(
n
)
. The
F
refers to the expression involving all these terms it's not the function that solves the equation.
Our text sometimes uses what may seem to be a di erent general expression,
d
n
y
dx
n
=
f
(
x, y, y
0
, ..., y
(
n

1)
)
,
but really that's the same sort of equation, but solved for
y
(
n
)
. Since if
y
(
n
)
occurs nontrivially (not multiplied
by 0, which
would
be pretty trivial), it
can
be solved for, at least in theory.
The point is to nd a
solution
of the equation, which is a function
y
=
y
(
x
)
which
ts into the equation.
For example, you can check that the function:
y
=

e
x
/
11
does solve the equation in the previous paragraph.
However, that isn't the only function that solves that di erential equation. The functions:
y
=
2
e
((
√
93

5)
x/
2)

e
x
/
11
and
y
=
3
e
((

√
93

5)
x/
2)

e
x
/
11
also solve the equation. Don't despair at those stupid
√
93
's. You'll see, soon, how those got there. And.
you'll see that it wasn't hard to nd those solutions, how they are all related, and why I knew how many
di erent solutions there might be. For now, you should be able at least to check that the functions do indeed
solve the equation.
There is a dirty little secret here. I didn't really solve that equation by hand. I used a computeralgebra