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Exam III Review Sheet
Math 2065
The syllabus for Exam III is Sections 4.6, 5.1{5.5, the matrix algebra supplement, and 7.1{7.3.
You should review the assigned exercises in these sections. Following is a brief list (not necessarily
complete) of terms, skills, and formulas with which you should be familiar.
²
If
f
y
1
(
t
)
; y
2
(
t
)
g
is a fundamental solution set for the homogeneous equation
y
00
+
b
(
t
)
y
0
+
c
(
t
)
y
= 0
;
know how to use the method of
variation of parameters
to ﬂnd a particular solution
y
p
of the
nonhomogeneous equation
y
00
+
b
(
t
)
y
0
+
c
(
t
)
y
=
f
(
t
)
:
In this method, it is not necessary for
f
(
t
) to be an exponential polynomial.
Variation of Parameters:
Find
y
p
in the form
y
p
=
u
1
y
1
+
u
2
y
2
where
u
1
and
u
2
are
unknown functions whose derivatives satisfy the following two equations:
(
/
)
u
0
1
y
1
+
u
0
2
y
2
=
0
u
0
1
y
0
1
+
u
0
2
y
0
2
=
f
(
t
)
:
Solve the system (
/
) for
u
0
1
and
u
0
2
, and then integrate to ﬂnd
u
1
and
u
2
.
²
Know what it means for a function to have a
jump discontinuity
and to be
piecewise contin
uous
.
²
Know how to piece together solutions on diﬁerent intervals to produce a solution of one of
the initial value problems
y
0
+
ay
=
f
(
t
)
;
y
(
t
0
) =
y
0
;
or
y
00
+
ay
0
+
by
=
f
(
t
)
;
y
(
t
0
) =
y
0
; y
0
(
t
0
) =
y
1
;
where
f
(
t
) is a piecewise continuous function on an interval containing
t
0
.
²
Know what the
unit step function
(also called the
Heaviside function
) (
h
(
t
¡
c
)) and the
onoﬁ switches
(
´
[
a;b
)
) are:
h
(
t
¡
c
) =
h
c
(
t
) =
(
0 if 0
•
t < c;
1 if
c
‚
t
and,
´
[
a;b
)
=
(
1 if
a
•
t < b;
0 otherwise
=
h
(
t
¡
a
)
¡
h
(
t
¡
b
)
;
and know how to use these two functions to rewrite a piecewise continuous function in a
manner which is convenient for computation of Laplace transforms.
²
Know the second translation principle (Theorem 8.2.4):
Lf
f
(
t
¡
c
)
h
(
t
¡
c
)
g
=
e
¡
cs
F
(
s
)
and how to use it (particulary in the form of Corollary 8.2.5):
Lf
g
(
t
)
h
(
t
¡
c
)
g
=
e
¡
cs
Lf
g
(
t
+
c
)
g
as a tool for calculating the Laplace transform of piecewise continuous functions.
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 Winter '09

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