Chap 1 1st order differential equations
1. Introduction
2. Separable differential equations 3. Linear differential equations
4. Exact differential equations 5. Special differential equations
It should be completely understood that anything with the possibility of violating the law of copyright should not be cited in this lecture note. All the illustrative figures and/or numerical examples must be referred to the assigned textbook.
A differential equation consists of
an unknown function y, an independent variable x, and the derivative of the function y
(n)(x)
n n
n
dx x y x d
y ( )
) (
where
( )=
y
(n)(x)=f(x,y, y’, …,y
(n-1)) n-th order
y
’(x)=f(x,y) 1st order
A separable differential equation:
y’(x)=A(x)B(y)
0 )
( if
) ) (
( = ≠
⇒ A x dx B y
y B
dy
A linear differential equation:
y’(x)+p(x)y(x)=q(x)
∫ ∫ +
∫ =
⇒
= ∫
⎟ ⎠
⎜ ⎞
⎝
⇒ ⎛ ∫
= ∫
C dx
qe ye
qe dx ye
d
e
dx x p dx
x p
dx x p dx
x p
dx x p
) ( )
(
) ( )
(
)
I.F.
( Integration FactorAn exact differential equation:
M(x,y) y’+N(x,y)=0
C y
x
x y x N y
y x M
y y y x
x x N
y y x
x M
⇒ =
∂
= ∂
∂
⇒ ∂
∂
= ∂
∂
= ∂
) ,
(
) ,
( )
, (
) ,
) ( ,
) ( ,
) ( ,
(
ϕ
ϕ ϕ
Potential function φ(x,y)
A nonexact differential equation:
M(x,y) y’+N(x,y)=0
C y
x
y y y x
x x N
y y x
x M
x
y x N y
y x M
x y x N y
y x M
⇒ =
∂
= ∂
∂
= ∂
⇒
∂
= ∂
∂
⇒ ∂
∂
≠ ∂
∂
∂
) ,
(
) ,
) ( ,
) ( ,
) ( ,
(
)]
, ( [
)]
, ( [
) ,
( )
, (
ϕ
μ ϕ μ ϕ
μ
μ
Special differential equations:
(1) Homogeneous y’(x)=f(y/x)
x dx u
u f
dy
xu u
y
x u y
− =
⇒
+
⇒ =
=
) (
'
'
Special differential equations:
(2) Bernoulli
y’(x)+p(x)y=q(x)y
a) ( )
1 ( '
' )
1 ( '
1
x q u
x a p
u
y y
a u
y u
a a
=
− +
⇒
−
⇒ =
=
−
−
Special differential equations:
(3) Riccatti
y’(x)=p(x)y
2+q(x)y+r(x)
0 )
2 ( '
1 ) (
1 ) ' (
' '
1
2 2
= +
+
⇒ +
+ +
+ +
=
−
⇒ =
+
=
p z
q ps
z
z r s
z q s
z p s z
y
s z
y
Uniqueness of the initial value problem :
y’(x)=f(x,y) y(x
o)=y
o→ f(x,y) is continuous on R centered at (x
o,y
o)
is continuous on R centered at (x y
o,y
o)
f
∂
∂
Chap 2 2nd order differential equations
1. Introduction
2. Order reduction
3. Homogeneous differential equations
4. Nonhomogeneous differential equations
It should be completely understood that anything with the possibility of violating the law of copyright should not be cited in this lecture note. All the illustrative figures and/or numerical examples must be referred to the assigned textbook.
A differential equation consists of
an unknown function y, an independent variable x, and the highest derivative is y
(2)(x)
y
(2)(x)=f(x,y, y’) 2nd order
y
’(x)=f(x,y) 1st order
A homogeneous differential equation:
y’’(x) +A(x)y’(x)+B(x)y(x)=0
) (
) (
) (
) (
) (
2 1
2 1
x cy
x y
x by
x ay
x y
≠
+
⇒ =
A homogeneous differential equation with constant coefficients
y’’(x) +Ay’(x)+By(x)=0
2 , 4
0
2 2
1
2
B A
A
B A
e y
x−
±
= −
⇒
= +
+
→
=
λ λ
λ
λ
λ
y’’(x) +Ay’(x)+By(x)=0
) (
) (
) (
sin )
(
cos )
(
1 ,
4
) (
) (
4
) (
) (
4
2 1
2 1
2 1
2
2 1
2
2 1
2
1 1
2 1
x by
x ay
x y
qx e
x y
qx e
x y
i iq
p B
A
xe x
y e
x y
B A
e x
y e
x y
B A
px px
x x
x x
+
⇒ =
=
=
−
=
±
=
→
<
=
=
→
=
=
=
→
>
λ λ
λ λ
λ
1. λ
2.
3.
A nonhomogeneous differential equation y’’(x) +Ay’(x)+By(x)=f(x)
→ y
p’’(x) +Ay
p’(x)+By
p(x)=f(x)
→ y(x) =ay
1(x)+by
2(x)+y
p(x)
A nonhomogeneous differential equation y’’(x) +p(x)y’(x)+q(x)y(x)=f(x)
→ y
p(x) =uy
1(x)+vy
2(x)
) ' (
0 )
( ' 1
) ' (
0 )
( ' 1
' ) '
(
1 1
1
2 2
2 2 1
2 1
x W
fy f
y y x
v W
x W
fy y
f
y x
u W
y y
y x y
W
=
=
= −
=
=
A nonhomogeneous differential equation y’’(x) +Ay’(x)+By(x)=f(x)
→ f(x) y
p(x)
qx e
b qx
e a
qx be
qx e
b qx
e a qx
ae
x b
e x
a e
e d
cx bx
ae
x b
x a
px px
x
px px
px
N
m
m m
x N
n
n n
x
x x
N
m
m m
N
n
n n
sin 2
cos 2
sin
sin 1
cos 1
cos
) (
2 0 0
2 2 0 0
+ + +
+
∑
∑
∑
∑
+
=
=
+
=
=
λ
λ λ
λ λ
A nonhomogeneous differential equation y’’(x) +Ay’(x)+By(x)=f(x)
If f(x)=
y
p(x)=
( )
( )
( e qx ) c x ( e qx )
x b
qx e
x a
qx e
x a
px m
m px
N
m
m m
px N
n
n n
px N
n
n n
sin cos
sin or
cos
2 0 0
0
∑ +
∑
∑
+
=
=
=
Euler equation:
x
2y’’(x) +Axy’(x)+By(x)=0
0 )
( )
( ' ) 1 (
) ( ''
)) (
' )
( '' 1 (
) (
''
) ( 1 '
) ( '
) ( )
(
2
= +
− +
−
=
=
=
→
=
t BY t
Y A
t Y
t Y
t x Y
x y
t x Y
x y
t Y x
y e
x
tConstant linear 2nd differential equations
) (
) (
) (
ln sin
) (
ln cos
) (
ln )
( )
(
) (
) (
sin )
( cos
) (
) ( )
(
) ( )
(
2 1
2 1
2 1
2 1
2 1
2 1
2 1
1 1
2 1
2 1
2 1
x by
x ay
x y
x q
x x
y x
q x
x y
x x
x y
x x
y
x x
y x
x y
qt e
t Y
qt e
t Y
te t
Y e
t Y
e t
Y e
t Y
p p
pt pt
t t
t t
+
⇒ =
=
=
=
=
=
=
=
=
=
=
=
=
λ λ
λ λ
λ λ
λ λ
Euler equation: x
2y’’(x) +Axy’(x)+By(x)=0
) (
) (
) (
) (
0
) (
) (
) (
2 2
1 1
0 1
2 2
2
1 1 2
x y
b x
y b
x y
b x
y
a a
a a
e x
y e
x y
e x
y
n n
n n
x n
x
x n
+ +
+
⇒ =
= +
+ +
+
=
=
=
L L
K
λ λ
λ
λ λ
λ
High order differential equation: y
(n)↔λ
na
ny
(n)(x)+…+a
2y’’(x) +a
1y’(x)+a
0y(x)=0
Chap 4 Series solution
1. Power series solution 2. Recurrence realtion 3. Frobenius series
It should be completely understood that anything with the possibility of violating the law of copyright should not be cited in this lecture note. All the illustrative figures and/or numerical examples must be referred to the assigned textbook.
Power series about x
0!
) (
)
! (
) (
) (
) (
0 )
( 0
0 0 )
( 0
0
n x a y
x n x
x y
x x
a x
y
n n
n
n n
n
n n
=
→
−
=
−
=
∑
∑
∞
=
∞
=
Taylor’s series about x
0Indicial equation and recurrence relation
∑
∑
∑
∞
=
−
∞
=
−
∞
=
−
−
=
−
=
−
=
2
2 0
1
1 0
0
0
) (
) 1 (
) (
''
) (
) (
'
) (
) (
n
n n
n
n n
n
n n
x x
a n
n x
y
x x
na x
y
x x
a x
y
∑
∑
∑
∑
∞
= +
∞
=
−
∞
= +
∞
=
−
− +
+
=
−
−
=
− +
=
−
=
0
0 2
2
2 0
0
0 1
1
1 0
) (
) 1 )(
2 (
) (
) 1 (
) (
''
) (
) 1 (
) (
) (
'
n
n n
n
n n
n
n n
n
n n
x x
a n
n
x x
a n
n x
y
x x
a n
x x
na x
y
Frobenius series
∑
∑
∑
∞
=
− +
∞
=
− +
∞
=
+
−
−
=
− +
=
−
=
0
2 0
0
1 0
0
0
) (
) 1 (
) (
''
) (
) (
) (
'
) (
) (
n
r n n
n
r n n
n
r n n
x x
c n
n x
y
x x
c r
n x
y
x x
c x
y
∑
∑
∑
∑
∞
−
=
+ +
∞
=
− +
∞
−
=
+ +
∞
=
− +
− +
+ +
+
=
−
− +
+
=
− +
+
=
− +
=
2
0 2
0
2 0
1
0 1
0
1 0
) (
) 1 )(
2 (
) (
) 1 )(
( )
( ''
) (
) 1 (
) (
) (
) (
'
n
r n n
n
r n n
n
r n n
n
r n n
x x
c r
n r
n
x x
c r
n r
n x
y
x x
c r
n
x x
c r
n x
y
Chap 3 Laplace transform
1. Laplace transform
2. Discontinuous functions 3. Convolution
4. Solve differential equations by the Laplace transform
It should be completely understood that anything with the possibility of violating the law of copyright should not be cited in this lecture note. All the illustrative figures and/or numerical examples must be referred to the assigned textbook.
Laplace transform L{f(t)}=F(s)
) (
) (
)}
( )
( {
) (
)}
( {
) (
) (
)}
(
{
0s bG
s aF
t bg
t af
t f
s F
dt e
t f
s F
t
f
st+
= +
=
=
= ∫∞ −
L L L
1 -
Inverse Laplace transform
If L{f(t)}=F(s)
⎩ ⎨
⎧
≥
= <
−
=
−
−
−
=
−
a t
a a t
t H
s F
e t
H t
f
a s
F t
f e
s at
1 ) 0
(
) (
)}
( )
( {
) (
)}
( {
τ τ
τ
L L
Heaviside function (step function)
If L{f(t)}=F(s)
) (
) 1 (
)}
( {
) 1 (
} )
( {
) 0 ( )
0 ( '
) 0 ( )
( )}
( {
) (
) 1 (
2
1 )
(
s F
t t
s s F
dt t
f
f f
s
f s
s F
s t
f
n n
n
n n
n n
n
−
=
=
−
−
−
=
∫
−
−
−
L L L
L
Convolution f(t)*g(t)=g(t)*f(t)
) (
) (
)}
(
* ) (
{
) (
* ) (
) (
) (
) (
) (
) (
* ) (
0 0
s G
s F
t g
t f
t f
t g
d g
t f
d t
g f
t g
t f
t t
=
=
−
=
−
=
∫
∫
L
τ τ
τ
τ τ
τ
Dirac delta function δ(t-a)
[ ]
1 )}
( {
)}
( {
) ( )
( )
(
) (
) 1 (
) (
0
lim
0=
=
−
=
−
−
−
−
−
=
−
−
∞
→
∫
+
t
e a
t
f dt
t t
f
a t
H a
t H a
t
as
δ δ
τ τ
δ ε ε δ
ε
L
L
Chap 13 Fourier’s Series
1. Fourier’s Series
2. Fourier cos/sin Series
3. Complex Fourier’s Series
It should be completely understood that anything with the possibility of violating the law of copyright should not be cited in this lecture note. All the illustrative figures and/or numerical examples must be referred to the assigned textbook.
Fourier series (F.S.) of f(x), |x|≦L
∫
∫
∑
−
−
∞
=
=
=
→
+ +
=
L n L
L n L
n n
n
L dx x x n
L f b
L dx x x n
L f a
L x b n
L x a n
S a F
π π
π π
sin )
1 (
cos )
1 (
sin 2 cos
. .
1 0
Convergence of the Fourier series of f(x)
)]
( )
( 2 [
) 1 (
of .
. ) (
)]
( )
( 2 [
) 1 (
of .
. ) (
+
−
+
−
− +
→
±
=
+
→
<
<
−
L f
L f
x f
S F
L x
ii
x f
x f
x f
S F
L x
L
i
Fourier cosine series (F.C.S.) of f(x), 0≦x≦L
∫
∑
=
→
=
∞=
L n
n
n
L dx x x n
L f a
L x a n
S C
F
0 0
cos )
2 (
cos .
. .
π
π
Convergence of the Fourier cosine series of f(x)
) (
) (
of .
. .
) (
) 0
( )
( of
. .
. 0
) (
)]
( )
( 2 [
) 1 (
of .
. .
0 )
(
− + +
−
→
=
→
=
+
→
<
<
L f
x f
S C
F L
x iii
f x
f S
C F
x ii
x f
x f
x f
S C
F
L x
i
Fourier sine series (F.S.S.) of f(x), 0≦x≦L
∫
∑
=
→
=
∞=
L n
n
n
L dx x x n
L f b
L x b n
S S
F
0 1
sin )
2 (
sin .
. .
π
π
Convergence of the Fourier sine series of f(x)
0 )
( of
. .
. )
(
0 )
( of
. .
. 0
) (
)]
( )
( 2 [
) 1 (
of .
. .
0 )
(
→
=
→
=
+
→
<
<
+
−
x f
S C
F L
x iii
x f
S C
F x
ii
x f
x f
x f
S S
F
L x
i
Complex Fourier series (CFS) of f(x)=f(x+p) where p is the fundamental period
p
dx e
x p f
d
e d
S F
C
p a
a
x in
n
n
x in
n
ω π
ω ω
2
) 1 (
. .
.
0
0 0
=
=
→
=
∫
∑
+ −
∞
−∞
=
Chap 14 Fourier’s transform
1. Fourier’s transform
2. Fourier’s cos/sin transform
3. Finite Fourier’s cos/sin transform
It should be completely understood that anything with the possibility of violating the law of copyright should not be cited in this lecture note. All the illustrative figures and/or numerical examples must be referred to the assigned textbook.
Fourier transform F{f(t)}=F( ω )
) (
) (
)}
( )
( {
) 2 (
) 1 (
)}
( {
) ( )
( )}
( {
ω ω
ω π ω
ω
ω
ω ω
bG aF
t bg t
af
d e
F t
f F
dt e
t f
F t
f
t i t
i
+
= +
=
=
=
=
∫
∫
∞
∞
−
∞
∞
−
−
F F F
1 -
Inverse Fourier transform
If F{f(t)}=F( ω )
) 2 (
)} 1 (
{
)
| (
| )} 1
( {
) (
)}
( )
( {
) (
)}
( {
π ω
ω
ω τ
τ
ω
τω
f t
F
F a at a
f
F e
t H
t f
a F
t f
e
i iat
=
=
=
−
−
−
=
−
F
F
F
F
If F{f(t)}=F( ω )
) (
) ( )}
( {
) 1 (
} )
( {
) (
) (
)}
( {
) ( )
(
ω ω ω
ω ω
n n
n
n n
F i
t t
i F dt
t f
F i
t f
=
=
=
∫
F
F
F
Convolution f(t)*g(t)=g(t)*f(t)
) (
) (
)}
(
* ) (
{
) (
* ) (
) (
) (
) (
) (
) (
* ) (
ω ω
τ τ
τ
τ τ
τ
G F
t g
t f
t f
t g
d g
t f
d t
g f
t g
t f
=
=
−
=
−
=
∫
∫
∞
∞
−
∞
∞
−
F
Dirac delta function δ(t-a)
[ ]
1 )}
( {
1 )}
( {
)}
( {
) ( )
( )
(
) (
) 1 (
) (
0
lim
0=
=
=
−
=
−
−
−
−
−
=
−
−
∞
→
∫
+
t t
e a
t
f dt
t t
f
a t
H a
t H a
t
ia
δ δ
δ
τ τ
δ ε ε δ
ω ε
L F
F
Fourier cosine transform F
c{f(t)}=F
c( ω )
∫
∫
∞
∞
=
=
=
=
0 0
cos )
2 ( ) 1
( )}
( {
cos )
( )
( )}
( {
ω ω
π ω ω
ω ω
td F
t f
F
tdt t
f F
t f
c
c
1 - c c
F F
Inverse Fourier cosine transform
Fourier sine transform F
s{f(t)}=F
s( ω )
∫
∫
∞
∞
=
=
=
=
0 0
sin )
2 ( ) 1
( )}
( {
sin )
( )
( )}
( {
ω ω
π ω ω
ω ω
td F
t f
F
tdt t
f F
t f
s
s
1 - s s
F F
Inverse Fourier sine transform
Finite Fourier cosine transform F
cn{f(t)}=F
cn(n)
∑
∫
∞
=
=
=
=
=
0 0
cos )
2 ( )
( )}
( {
cos )
( )
( )}
( {
n
cn cn
cn
nt n
F t
f n
F
ntdt t
f n
F t
f
π
π
1 - cn cn
F F
Inverse finite Fourier cosine transform
Finite Fourier sine transform F
sn{f(t)}=F
sn(n)
∑
∫
∞
=
=
=
=
=
1 0
sin )
2 ( )
( )}
( {
sin )
( )
( )}
( {
n
sn sn
sn
nt n
F t
f n
F
ntdt t
f n
F t
f
π
π
1 - sn sn
