A-56
Differential Equations
Differential equation
Method of solution
Euler or Cauchy equation
x
2
d
2
y
dx +
bx
dy
dx +
cy = S(x)
Putting x = e
t
, the equation becomes
d
2
y
dt
2
+ (b − 1)
dy
dt +
cy = S(e
t
)
and can then be solved as a linear second order
equation.
Bessel’s equation
x
2
d
2
y
dx
2
+ x
dy
dx +
(λ
2
x
2
− n
2
)y = 0
y = c
1
J
n
(λx) + c
2
Y
n
(λx)
Transformed Bessel’s equation
x
2
d
2
y
dx
2
+ (2p + 1)x
dy
dx +
�
α
2
x
2r
+ β
2
�
y = 0
y = x
−p
�
c
1
J
q/r
� α
r
x
r
�
+ c
2
Y
q/r
� α
r
x
r
��
where q =
�
p
2
− β
2
.
Legendre’s equation
(1 − x
2
)
d
2
y
dx
2
− 2x
dy
dx +
n(n + 1)y = 0
y = c
1
P
n
(x) + c
2
Q
n
(x)
Separation of variables
f
1
(x)g
1
(y) dx + f
2
(x)g
2
(y) dy = 0
� f
1
(x)
f
2
(x)
dx +
� g
2
(y)
g
1
(y)
dy = c
Exact equation
M(x, y) dx + N(x, y) dy = 0
where ∂ M/∂y = ∂ N/∂x
�
M∂x +
� �
n −
∂
∂
y
�
M∂x
�
dy = c where ∂x indicates
that the integration is to be performed with respect to x
keeping y constant.
Linear first order equation
dy
dx +
P(x)y = Q(x)
ye
�
P dx
=
�
Qe
�
P dx
dx + c
Bernoulli’s equation
dy
dx +
P(x)y = Q(x)y
n
ve
(1−n)
�
P dx =
�
Qe
(1−n)
�
P dx
dx + c
where v = y
1−n
.
If n = 1, then the solution is ln y =
�
(Q− P) dx + c.
Homogeneous equation
dy
dx =
F
� y
x
�
ln x =
�
dv
F (v)−v
+ c where v = y/x.
If F (v) = v, then the solution is y = cx.
Reducible to homogeneous
(a
1
x + b
1
y + c
1
) dx
+(a
2
x + b
2
y + c
2
) dy = 0
with
a
1
a
2
�=
b
1
b
2
Set u = a
1
x + b
1
y + c
1
and v = a
2
x + b
2
y + c
2
. Then
eliminate x and y and the equation becomes
homogenous.
Reducible to separable
(a
1
x + b
1
y + c
1
) dx
+(a
2
x + b
2
y + c
2
) dy = 0
with
a
1
a
2
=
b
1
b
2
Set u = a
1
x + b
1
y. Then eliminate x or y and the
equation becomes separable.
FOURIER SERIES
1. If f (x) is a bounded periodic function of period 2L (i.e., f (x + 2L) = f (x)), and satisfies the Dirichlet conditions:
(a) In any period f (x) is continuous, except possibly for a finite number of jump discontinuities.
(b) In any period f (x) has only a finite number of maxima and minima.
Then f (x) may be represented by the Fourier series
a
0
2 +
∞
�
n=1
�
a
n
cos
nπ x
L +
b
n
sin
nπ x
L
�
where a
n
and b
n
are as determined below. This series will converge to f (x) at every point where f (x) is continuous, and to
f (x
+
) + f (x
−
)
2
(i.e., the average of the left-hand and right-hand limits) at every point where f (x) has a jump discontinuity.
a
n
=
1
L
�
L
−L
f (x) cos
nπ x
L
dx,
n = 0, 1, 2, 3, . . . ,
b
n
=
1
L
�
L
−L
f (x) sin
nπ x
L
dx,
n = 1, 2, 3, . . .
We may also write
a
n
=
1
L
�
α
+2L
α
f (x) cos
nπ x
L
dx and b
n
=
1
L
�
α
+2L
α
f (x) sin
nπ x
L
dx
where α is any real number. Thus if α = 0,
a
n
=
1
L
�
2L
0
f (x) cos
nπ x
L
dx,
n = 0, 1, 2, 3, . . . ,
b
n
=
1
L
�
2L
0
f (x) sin
nπ x
L
d,
n = 1, 2, 3, . . .
2. If in addition to the restrictions in (1), f (x) is an even function (i.e., f (−x) = f (x)), then the Fourier series reduces to
a
0
2 +
∞
�
n=1
a
n
cos
nπ x
L
That is, b
n
= 0. In this case, a simpler formula for a
n
is
a
n
=
2
L
�
L
0
f (x) cos
nπ x
L
dx,
n = 0, 1, 2, 3, . . .
3. If in addition to the restrictions in (1), f (x) is an odd function (i.e., f (−x) = − f (x)), then the Fourier series reduces to
∞
�
n=1
b
n
sin
nπ x
L
That is, a
n
= 0. In this case, a simpler formula for the b
n
is
b
n
=
2
L
�
L
0
f (x) sin
nπ x
L
dx,
n = 1, 2, 3, . . .
4. If in addition to the restrictions in (2) above, f (x) = − f (L − x), then a
n
will be 0 for all even values of n, including n = 0.
Thus in this case, the expansion reduces to
∞
�
m=1
a
2m−1
cos
(2m − 1)πx
L
A-57
A-58
Fourier Series
5. If in addition to the restrictions in (3) above, f (x) = f (L − x), then b
n
will be 0 for all even values of n. Thus in this case,
the expansion reduces to
∞
�
m=1
b
2m−1
sin
(2m − 1)πx
L
(The series in (4) and (5) are known as odd-harmonic series, since only the odd harmonics appear. Similar rules may be stated
for even-harmonic series, but when a series appears in the even-harmonic form, it means that 2L has not been taken as the
smallest period of f (x). Since any integral multiple of a period is also a period, series obtained in this way will also work,
but in general computation is simplified if 2L is taken to be the smallest period.)
6. If we write the Euler definitions for cos θ and sin θ, we obtain the complex form of the Fourier series known either as the
“Complex Fourier Series” or the “Exponential Fourier Series” of f (x). It is represented as
f (x) =
1
2
n=+∞
�
n=−∞
c
n
e
iω
n
x
where
c
n
=
1
L
�
L
−L
f (x) e
−iω
n
x
dx,
n = 0, ±1, ±2, ±3, . . .
with ω
n
=
nπ
L
for n = 0, ±1, ±2, . . . The set of coefficients c
n
is often referred to as the Fourier spectrum.
7. If both sine and cosine terms are present and if f (x) is of period 2L and expandable by a Fourier series, it can be represented
as
f (x) =
a
0
2 +
∞
�
n=1
c
n
sin
� nπx
L +
φ
n
�
,
where
a
n
= c
n
sin φ
n
,
b
n
= c
n
cos φ
n
,
c
n
=
�
a
2
n
+ b
2
n
,
φ
n
= arctan
� a
n
b
n
�
It can also be represented as
f (x) =
a
0
2 +
∞
�
n=1
c
n
cos
� nπx
L +
φ
n
�
,
where
a
n
= c
n
cos φ
n
,
b
n
= −c
n
sin φ
n
,
c
n
=
�
a
2
n
+ b
2
n
,
φ
n
= arctan
�
−
b
n
a
n
�
where φ
n
is chosen so as to make a
n
, b
n
, and c
n
hold.
8. The following table of trigonometric identities should be helpful for developing Fourier series.
n
n even
nodd
n/2 odd
n/2 even
sin nπ
0
0
0
0
0
cos nπ
(−1)
n
+1
−1
+1
+1
∗ sin
nπ
2
0
(−1)
(n−1)/2
0
0
∗ cos
nπ
2
(−1)
n/2
0
−1
+1
sin
nπ
4
√
2
2
(−1)
(n
2
+4n+11)/8
(−1)
(n−2)/4
0
*A useful formula for sin
nπ
2
and cos
nπ
2
is given by
sin
nπ
2 =
(i)
n+1
2
[(−1)
n
− 1] and cos
nπ
2 =
(i)
n
2
[(−1)
n
+ 1],
where i
2
= −1.
Auxiliary Formulas for Fourier Series
1 =
4
π
�
sin
π
x
k +
1
3
sin
3π x
k +
1
5
sin
5π x
k + · · ·
�
[0 < x < k]
x =
2k
π
�
sin
π
x
k −
1
2
sin
2π x
k +
1
3
sin
3π x
k − · · ·
�
[−k < x < k]
x =
k
2 −
4k
π
2
�
cos
π
x
k +
1
3
2
cos
3π x
k +
1
5
2
cos
5π x
k + · · ·
�
[0 < x < k]
x
2
=
2k
2
π
3
��
π
2
1 −
4
1
�
sin
π
x
k −
π
2
2
sin
2π x
k +
�
π
2
3 −
4
3
3
�
sin
3π x
k
−
π
2
4
sin
4π x
k +
�
π
2
5 −
4
5
3
�
sin
5π x
k + · · ·
�
[0 < x < k]
x
2
=
k
2
3 −
4k
2
π
2
�
cos
π
x
k −
1
2
2
cos
2π x
k +
1
3
2
cos
3π x
k −
1
4
2
cos
4π x
k + · · ·
�
[−k < x < k]
1 −
1
3 +
1
5 −
1
7 + · · · =
π
4
1 −
1
2
2
+
1
3
2
+
1
4
2
+ · · · =
π
2
6
1 −
1
2
2
+
1
3
2
−
1
4
2
+ · · · =
π
2
12
1 +
1
3
2
+
1
5
2
−
1
7
2
+ · · · =
π
2
8
1
2
2
+
1
4
2
+
1
6
2
+
1
8
2
+ · · · =
π
2
24
FOURIER EXPANSIONS FOR BASIC PERIODIC FUNCTIONS
f (x) =
4
π
�
n=1,3,5...
1
n
sin
nπ x
L
f (x) =
2
π
∞
�
n=1
(−1)
n
n
�
cos
nπc
L
− 1
�
sin
nπ x
L
f (x)=
c
L
+
2
π
∞
�
n=1
(−1)n
n
sin
nπc
L
cos
nπ x
L
f (x) =
2
L
∞
�
n=1
sin
nπ
2
sin(
1
2
nπc/L)
1
2
nπc/L
sin
nπ x
L
A-59
A-58
Fourier Series
5. If in addition to the restrictions in (3) above, f (x) = f (L − x), then b
n
will be 0 for all even values of n. Thus in this case,
the expansion reduces to
∞
�
m=1
b
2m−1
sin
(2m − 1)πx
L
(The series in (4) and (5) are known as odd-harmonic series, since only the odd harmonics appear. Similar rules may be stated
for even-harmonic series, but when a series appears in the even-harmonic form, it means that 2L has not been taken as the
smallest period of f (x). Since any integral multiple of a period is also a period, series obtained in this way will also work,
but in general computation is simplified if 2L is taken to be the smallest period.)
6. If we write the Euler definitions for cos θ and sin θ, we obtain the complex form of the Fourier series known either as the
“Complex Fourier Series” or the “Exponential Fourier Series” of f (x). It is represented as
f (x) =
1
2
n=+∞
�
n=−∞
c
n
e
iω
n
x
where
c
n
=
1
L
�
L
−L
f (x) e
−iω
n
x
dx,
n = 0, ±1, ±2, ±3, . . .
with ω
n
=
nπ
L
for n = 0, ±1, ±2, . . . The set of coefficients c
n
is often referred to as the Fourier spectrum.
7. If both sine and cosine terms are present and if f (x) is of period 2L and expandable by a Fourier series, it can be represented
as
f (x) =
a
0
2 +
∞
�
n=1
c
n
sin
� nπx
L +
φ
n
�
,
where
a
n
= c
n
sin φ
n
,
b
n
= c
n
cos φ
n
,
c
n
=
�
a
2
n
+ b
2
n
,
φ
n
= arctan
� a
n
b
n
�
It can also be represented as
f (x) =
a
0
2 +
∞
�
n=1
c
n
cos
� nπx
L +
φ
n
�
,
where
a
n
= c
n
cos φ
n
,
b
n
= −c
n
sin φ
n
,
c
n
=
�
a
2
n
+ b
2
n
,
φ
n
= arctan
�
−
b
n
a
n
�
where φ
n
is chosen so as to make a
n
, b
n
, and c
n
hold.
8. The following table of trigonometric identities should be helpful for developing Fourier series.
n
n even
nodd
n/2 odd
n/2 even
sin nπ
0
0
0
0
0
cos nπ
(−1)
n
+1
−1
+1
+1
∗ sin
nπ
2
0
(−1)
(n−1)/2
0
0
∗ cos
nπ
2
(−1)
n/2
0
−1
+1
sin
nπ
4
√
2
2
(−1)
(n
2
+4n+11)/8
(−1)
(n−2)/4
0
*A useful formula for sin
nπ
2
and cos
nπ
2
is given by
sin
nπ
2 =
(i)
n+1
2
[(−1)
n
− 1] and cos
nπ
2 =
(i)
n
2
[(−1)
n
+ 1],
where i
2
= −1.
Auxiliary Formulas for Fourier Series
1 =
4
π
�
sin
π
x
k +
1
3
sin
3π x
k +
1
5
sin
5π x
k + · · ·
�
[0 < x < k]
x =
2k
π
�
sin
π
x
k −
1
2
sin
2π x
k +
1
3
sin
3π x
k − · · ·
�
[−k < x < k]
x =
k
2 −
4k
π
2
�
cos
π
x
k +
1
3
2
cos
3π x
k +
1
5
2
cos
5π x
k + · · ·
�
[0 < x < k]
x
2
=
2k
2
π
3
��
π
2
1 −
4
1
�
sin
π
x
k −
π
2
2
sin
2π x
k +
�
π
2
3 −
4
3
3
�
sin
3π x
k
−
π
2
4
sin
4π x
k +
�
π
2
5 −
4
5
3
�
sin
5π x
k + · · ·
�
[0 < x < k]
x
2
=
k
2
3 −
4k
2
π
2
�
cos
π
x
k −
1
2
2
cos
2π x
k +
1
3
2
cos
3π x
k −
1
4
2
cos
4π x
k + · · ·
�
[−k < x < k]
1 −
1
3 +
1
5 −
1
7 + · · · =
π
4
1 −
1
2
2
+
1
3
2
+
1
4
2
+ · · · =
π
2
6
1 −
1
2
2
+
1
3
2
−
1
4
2
+ · · · =
π
2
12
1 +
1
3
2
+
1
5
2
−
1
7
2
+ · · · =
π
2
8
1
2
2
+
1
4
2
+
1
6
2
+
1
8
2
+ · · · =
π
2
24
FOURIER EXPANSIONS FOR BASIC PERIODIC FUNCTIONS
f (x) =
4
π
�
n=1,3,5...
1
n
sin
nπ x
L
f (x) =
2
π
∞
�
n=1
(−1)
n
n
�
cos
nπc
L
− 1
�
sin
nπ x
L
f (x)=
c
L
+
2
π
∞
�
n=1
(−1)n
n
sin
nπc
L
cos
nπ x
L
f (x) =
2
L
∞
�
n=1
sin
nπ
2
sin(
1
2
nπc/L)
1
2
nπc/L
sin
nπ x
L
A-59