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Mathematics 33B: Milestones for Week 1
Book sections: 2.1, 2.2, 2.4, 2.5.
TOPIC 1:
Basic deﬁnitions
Read the ﬁrst chapter and know what is meant by the normal form of a
diﬀerential equation, an initial value problem, and the interval of existence.
Example: The equation
x
00
=

g
is in normal form.
Example:
±
x
00
=

g
x
0
(0) =
v
0
,
x
(0) =
x
0
Is an initial value problem and its solution is
x
(
t
) =
x
0
+
v
0
t

g
2
t
2
.
The
value of
x
(0) and
x
0
(0) are called initial values, they are several values given
at one point. Boundary values for example might be the values
x
(0) and
x
(
.
42), values given at two or more points. A problem where a diﬀerential
equation is given with boundary values is called a boundary value problem.
Example: Interval of existence. The book example is nice, we use it.
Consider the initial value problem
y
0
=
y
2
with
y
(0) = 1. A solution is

1
t

1
.
The interval of existence is (
∞
,
1).
TOPIC 2:
Direction Fields.
You should know how to geometrically interpret the solutions of a diﬀer
ential equation. We look at the following examples in class:
Example:
y
0
= 1. Solution:
y
(
t
) =
t
+
C.
Example: The exponential equation,
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 Spring '07
 staff
 Math

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