Math 421:01 and 02 — Spring 2008
Final Exam Review and Formula Sheet
Trigonometry
The formula
e
it
D
cos
t
C
i
sin
t
(with
i
2
D
1
) connects exponential and trigonometric functions. Consequences include the basic identities
cos
.˛
C
ˇ/
D
cos
˛
cos
ˇ
sin
˛
sin
ˇ
sin
.˛
C
ˇ/
D
sin
˛
cos
ˇ
C
cos
˛
sin
ˇ
that are used to prove the
orthogonality
of the functions
f
1; .
cos
nx;
sin
nx
W
n
D
1; 2; : : :/
g
on the interval
< x <
.
Laplace transforms
The
Laplace transform
of a function of
t
is a function of a new variable
s
defined by
ˇ
f
f .t/
g D
Z
1
0
f .t/e
st
dt
This definition shows that the Laplace transform is a
linear operator
.
In describing properties of the
Laplace transform, it is conventional to write
F.s/
for
ˇ
f
f .t/
g
.
Some useful properties (in addition to
linearity) are
f .t/
F.s/
1
1
s
t
n
nŠ
1
s
n
C
1
cos
t
s
s
2
C
1
sin
t
1
s
2
C
1
f
0
.t/
sF.s/
f .0/
f .bt/
1
b
F
.s=b/
e
at
f .t/
F.s
a/
tf .t/
F
0
.s/
f .t
a/
U
.t
a/
e
as
F.s/
ı.t
a/
e
as
.f
g/.t/
F.s/G.s/
In the last formula,
.f
g/.t/
denotes the
convolution
of
f .t/
and
g.t/
which is defined by
.f
g/.t/
D
Z
t
0
f . /g.t
/ d
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Math 421:01 and 02, May 1314, 2008, p. 2
Calculation of Laplace transforms or inverse transforms typically requires several steps taken from this
table. To apply these rules, functions must be found so that the given expression has the form shown in a
line of this table.
Ordinary Differential Equations
There are two families of differential equations that are easy to solve exactly
.
)
Linear equations with constant coefficients
. Usually, the solutions have the form
y
D
e
cx
.
.
)
Euler equations
, whose terms are combinations with constant coefficients of
x
k
d
k
y=dx
k
for several
integer
k
. Usually, the solutions have the form
y
D
x
c
.
To solve these equations,
try
the correct form of solution.
The equation reduces to an
algebraic
equation
in
c
. If a root
c
is complex, the exponential is usually converted to a form containing trigonometric
functions. For Euler equations, this also requires the identity
x
c
D
e
c
ln
x
.
If the algebraic equation has
repeated roots
, this method does not give enough solutions. If
c
is a
repeated root, a second solution is
xe
cx
in the constant coefficient case, or
x
c
ln
x
in the Euler equation
case. A full characterization of solutions will not be given here, since it will be easy to check any tentative
solution constructed from this principle.
Fourier series
The
Fourier series
of a function
f .x/
on the interval
p < x < p
is given by
a
0
2
C
1
X
n
D
1
a
n
cos
n
x
p
C
b
n
sin
n
x
p
where
a
0
D
1
p
Z
p
p
f .x/ dx
a
n
D
1
p
Z
p
p
f .x/
cos
n
x
p
dx
b
n
D
1
p
Z
p
p
f .x/
sin
n
x
p
dx
. /
The
constant term
a
0
=2
is the
average value
of
f .x/
, and
Z
p
p
cos
2
n
x
p
dx
D
p
Z
p
p
sin
2
n
x
p
dx
D
p
so that the Fourier series of
f .x/
D
1
,
f .x/
D
cos
n
x=p
or
f .x/
D
sin
n
x=p
is
a single term
identical to
f .x/
.
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 Spring '08
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 Fourier Series, Boundary value problem, Partial differential equation, Sturm–Liouville theory

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