王 俊 鑫 (Chun-Hsin Wang)
中華大學 資訊工程系
Fall 2002
Chap 1 First-Order
Differential Equations

Page 2
Outline
 Basic Concepts
 Separable Differential Equations
 substitution Methods
 Exact Differential Equations
 Integrating Factors
 Linear Differential Equations
 Bernoulli Equations

Page 3
Basic Concepts
 Differentiation
x
e
x
x
x
a
a
a
e
e
nx
x
a
a
x
x
x
x
n
n
log
)
(log
1
)
(ln
ln
)
(
)
(
)
( 1









 
x
x
x
x
x
x
x
x
x
x
x
x
x
x
cot
csc
)
(csc
tan
sec
)
(sec
csc
)
(cot
sec
)
(tan
sin
)
(cos
cos
)
(sin
2
2
















Page 4
Basic Concepts
 Differentiation
x
x
x
x
sinh
)
(cosh
cosh
)
(sinh




2
1
2
1
2
1
2
1
1
1
)
(cot
1
1
)
(tan
1
1
)
(cos
1
1
)
(sin
x
x
x
x
x
x
x
x



















Page 5
Basic Concepts
 Integration
c
a
a
dx
a
c
e
dx
e
c
x
dx
x
dx
x
c
n
x
dx
x
x
x
x
x
n
n

















ln
ln
1
1
1
1



















vdx
u
uv
dx
v
u
vdu
uv
udv
udx
c
cudx
vdx
udx
dx
v
u )
(

Page 6
Basic Concepts
 Integration
c
x
x
xdx
c
x
x
xdx
c
x
xdx
c
x
xdx
c
x
xdx
c
x
xdx






















cot
csc
ln
csc
tan
sec
ln
sec
sin
ln
cot
cos
ln
tan
sin
cos
cos
sin

Page 7
Basic Concepts
 Integration
c
a
x
dx
a
x
c
a
x
dx
a
x
c
a
x
dx
x
a
c
a
x
a
dx
a
x




















1
2
2
1
2
2
1
2
2
1
2
2
cosh
1
sinh
1
sin
1
tan
1
1

Page 8
Basic Concepts
 ODE vs. PDE
 Dependent Variables vs. Independent
Variables
 Order
 Linear vs. Nonlinear
 Solutions

Page 9
Basic Concepts
 Ordinary Differential Equations
 An unknown function (dependent variable) y
of one independent variable x
x
dx
dy
y cos



0
4 


 y
y
2
2
2
)
2
(
2 y
x
y
e
y
y
x x










Page 10
Basic Concepts
 Partial Differential Equations
 An unknown function (dependent variable)
z of two or more independent variables
(e.g. x and y)
y
x
x
z
4
6 



y
x
y
x
z





2
2

Page 11
Basic Concepts
 The order of a differential equation is
the order of the highest derivative that
appears in the equation.
0
)
( 2
2
3






 y
n
x
y
x
y
x Order 2
2
2
1
y
x
dx
dy

 Order 1
1
)
( 4
3
2
2

 y
dx
y
d
Order 2

Page 12
Basic Concept
 The first-order differential equation contain only y’
and may contain y and given function of x.
 A solution of a given first-order differential equation
(*) on some open interval a<x<b is a function
y=h(x) that has a derivative y’=h(x) and satisfies
(*) for all x in that interval.
)
,
(
'
0
)
'
,
,
(
y
x
F
y
y
y
x
F


or (*)

Page 13
Basic Concept
 Example : Verify the solution
x
2
y
2y
xy'



Page 14
Basic Concepts
 Explicit Solution
 Implicit Solution
)
(x
h
y 
0
)
,
( 
y
x
H

Page 15
Basic Concept
 General solution vs. Particular solution
 General solution
 arbitrary constant c
 Particular solution
 choose a specific c
,....
2
,
3
'






c
c
sinx
y
cosx
y

Page 16
Basic Concept
 Singular solutions
 Def : A differential equation may sometimes have an
additional solution that cannot be obtained from the
general solution and is then called a singular
solution.
 Example
The general solution : y=cx-c2
A singular solution : y=x2/4
0
' 

 y
xy
y'
2

Page 17
Basic Concepts
 General Solution
 Particular Solution for y(0)=2 (initial condition)
kt
ce
t
y 
)
(
kt
e
t
y 2
)
( 
ky
y 


Page 18
Basic Concept
 Def: A differential equation together
with an initial condition is called an
initial value problem
0
0)
(
),
,
(
' y
x
y
y
x
f
y 


Page 19
Separable Differential Equations
 Def: A first-order differential equation of
the form
is called a separable differential
equation
dx
x
f
dy
y
g
f(x)
g(y)y
)
(
)
(
'



Page 20
Separable Differential Equations
 Example :
Sol:
0
4
9 

 x
y
y

Page 21
Separable Differential Equations
 Example :
Sol:
2
1 y
y 



Page 22
Separable Differential Equations
 Example :
Sol:
ky
y 


Page 23
Separable Differential Equations
 Example :
Sol:
1
)
0
(
,
2 


 y
xy
y

Page 24
Separable Differential Equations
 Substitution Method:
A differential equation of the form
can be transformed into a separable
differential equation
)
(
x
y
g
y 


Page 25
Separable Differential Equations
 Substitution Method:
ux
y  u
x
u
y 



x
dx
u
u
g
du
u
u
g
x
u
u
g
u
x
u










)
(
)
(
)
(

Page 26
Separable Differential Equations
 Example :
Sol:
2
2
2 x
y
y
xy 


cx
y
x
x
c
x
y
x
c
u
c
x
c
x
u
x
dx
u
udu
u
u
u
x
u
y
x
x
y
xy
x
xy
y
y
x
y
y
xy








































2
2
2
2
1
1
2
2
2
2
2
2
1
1
1
ln
ln
)
1
ln(
1
2
)
1
(
2
1
)
(
2
1
2
2
2

Page 27
Separable Differential Equations
 Exercise 1
2
01
.
0
1 y
y 


2
/
xy
y 

y
y
y
x 

 2
2
)
2
(
,
0
' 


 y
y
xy

Page 28
Exact Differential Equations
 Def: A first-order differential equation of
the form
is said to be exact if
0
)
,
(
)
,
( 
 dy
y
x
N
dx
y
x
M
x
y
x
N
y
y
x
M



)
,
(
)
,
(

Page 29
Exact Differential Equations
 Proof:
0
)
,
(
)
,
(
0
)
,
(









dy
y
x
N
dx
y
x
M
dy
y
u
dx
x
u
y
x
du
x
y
x
N
y
y
x
M
y
x
y
x
u






 )
,
(
)
,
(
)
,
(

Page 30
Exact Differential Equations
 Example :
Sol:
0
)
3
(
)
3
( 3
2
2
3



 dy
y
y
x
dx
xy
x
Exact
xy
x
N
y
M
xy
x
y
y
x
xy
y
xy
x
,
6
6
3
6
3
3
2
2
3














Page 31
Exact Differential Equations
Sol:
)
(
2
3
4
1
)
(
)
3
(
)
(
2
2
4
2
3
y
k
y
x
x
y
k
dx
xy
x
y
k
Mdx
u










1
4
3
2
2
4
)
(
3
)
(
3
c
y
y
k
y
y
x
N
dy
y
dk
y
x
y
u











Page 32
Exact Differential Equations
Sol:
c
y
y
x
x
y
x
u 


 )
6
(
4
1
)
,
( 4
2
2
4

Page 33
Exact Differential Equations
 Example
3
)
0
(
0
)
sinh
(cos
)
cosh
(sin



y
dy
y
x
dx
y
x

Page 34
Non-Exactness
 Example : 0


 xdy
ydx

Page 35
Integrating Factor
 Def: A first-order differential equation of the form
is not exact, but it will be exact if multiplied by
F(x, y)
then F(x,y) is called an integrating factor of this
equation
0
)
,
(
)
,
( 
 dy
y
x
Q
dx
y
x
P
0
)
,
(
)
,
(
)
,
(
)
,
( 
 dy
y
x
Q
y
x
F
dx
y
x
P
y
x
F

Page 36
Exact Differential Equations
 How to find integrating factor
 Golden Rule
x
x
y
y FQ
Q
F
FP
P
F
Exact
x
FQ
y
FP
FQdy
FPdx












,
0
)
(
1
1
0
Let
x
y
x
y
Q
P
Q
dx
dF
F
FQ
Q
dx
dF
FP
P
F(x)
F









Page 37
Exact Differential Equations
 Example :
Sol:
0


 xdy
ydx
Exact
x
N
x
y
M
dy
x
dx
x
y
x
xdy
ydx
x
F
,
1
1
1
2
2
2
2














Page 38
Exact Differential Equations
Sol:
cx
y
c
x
y
x
y
d
dy
x
dx
x
y







 0
)
(
1
2

Page 39
Exact Differential Equations
 Example :
2
)
2
(
0
)
cos(
)
sin(
2 2
2




y
dy
y
xy
dx
y

Page 40
Exact Differential Equations
 Exercise 2
0
2 2

 dy
x
xydx 0
)
( 2
2





d
r
rdr
e
x
e
F
ydy
ydx 

 ,
0
cos
sin
b
a
y
x
F
xdy
b
ydx
a 



 ,
0
)
1
(
)
1
(
0
)
1
(
)
1
( 


 dy
x
dx
y

Page 41
Linear Differential Equations
 Def: A first-order differential equation is
said to be linear if it can be written
 If r(x) = 0, this equation is said to be
homogeneous
)
(
)
( x
r
y
x
p
y 



Page 42
Linear Differential Equations
 How to solve first-order linear homogeneous
ODE ?
Sol:
0
)
( 

 y
x
p
y




 















dx
x
p
c
dx
x
p
c
dx
x
p
ce
e
e
e
y
c
dx
x
p
y
dx
x
p
y
dy
y
x
p
dx
dy
)
(
)
(
)
(
1
1
1
)
(
ln
)
(
0
)
(

Page 43
Linear Differential Equations
 Example :
Sol:
0


 y
y
x
c
x
c
x
dx
dx
x
p
e
c
e
ce
ce
ce
ce
x
y
2
)
1
(
)
(
1
1
)
(












Page 44
Linear Differential Equations
 How to solve first-order linear nonhomogeneous
ODE ?
Sol:
)
(
)
( x
r
y
x
p
y 


)
(
))
(
)
(
(
)
(
1
1
0
))
(
)
(
(
)
(
)
(
x
p
x
r
y
x
p
y
Q
P
Q
dx
dF
F
dy
dx
x
r
y
x
p
x
r
y
x
p
dx
dy
x
y 














Page 45
Linear Differential Equations
Sol:


dx
x
p
e
x
F
)
(
)
(





 




















c
dx
r
e
e
x
y
c
dx
r
e
y
e
r
e
y
e
py
y
e
dx
x
p
dx
x
p
dx
x
p
dx
x
p
dx
x
p
dx
x
p
dx
x
p
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(

Page 46
Linear Differential Equations
 Example :
Sol:
x
e
y
y 2



 
 
x
x
x
x
x
x
x
x
dx
dx
dx
x
p
dx
x
p
e
ce
c
e
e
c
dx
e
e
e
c
dx
e
e
e
c
dx
r
e
e
x
y
2
2
2
)
1
(
)
1
(
)
(
)
(
)
(











 








 












Page 47
Linear Differential Equations
 Example :
)
2
cos
2
2
sin
3
(
2 x
x
e
y
y x
'




Page 48
Bernoulli, Jocob
Bernoulli, Jocob
1654-1705

Page 49
Linear Differential Equations
 Def: Bernoulli equations
 If a = 0, Bernoulli Eq. => First Order
Linear Eq.
 If a <> 0, let u = y1-a
a
y
x
g
y
x
p
y )
(
)
( 


g
a
pu
a
u )
1
(
)
1
( 





Page 50
Linear Differential Equations
 Example :
Sol:
2
By
Ay
y 



 
A
B
ce
u
y
A
B
ce
c
dx
e
A
B
e
c
dx
Be
e
u
B
Au
u
Ay
B
Ay
By
y
y
y
u
y
y
y
u
Ax
Ax
Ax
Ax
Ax
Ax
a














































1
1
)
( 1
2
2
2
1
2
1
1

Page 51
Linear Differential Equations
 Exercise 3
4


 y
y kx
e
ky
y 



2
2 y
y
y 


1



 xy
xy
y
)
2
(
,
sin
3 
y
x
y
y 



Page 52
Summary
可分離 Separable 
變換法 Substitution 
正合 Exact 
積分因子 Integrating Factor 
線性 Linear 
柏努利 Bernoulli 
dx
x
f
dy
y
g )
(
)
( 
dx
x
f
du
u
g )
(
)
( 
0
)
,
(
)
,
( 
 dy
y
x
N
dx
y
x
M
0

 FQdy
FPdx
)
(
)
( x
r
y
x
p
y 


a
y
x
g
y
x
p
y )
(
)
( 



Page 53
Orthogonal Trajectories of
Curves
 Angle of intersection of two curves is
defined to be the angle between the
tangents of the curves at the point of
intersection
 How to use differential equations for
finding curves that intersect given
curves at right angles ?

Page 54
How to find Orthogonal Trajectories
 1st Step: find a differential equation
for a given cure
 2nd Step: the differential equation of the
orthogonal trajectories to be found
 3rd step: solve the differential equation
as above ( in 2nd step)
)
,
( y
x
f
y 
)
,
( y
x
f
y' 
)
,
(
1
y
x
f
y' 


Page 55
Orthogonal Trajectories of Curves
 Example: given a curve y=cx2, where c
is arbitrary. Find their orthogonal
trajectories.
Sol:

Page 56
Existance and Uniqueness of Solution
 An initial value problem may have no
solutions, precisely one solution, or
more than one solution.
 Example
1
)
0
(
,
0
' 

 y
y
y
1
)
0
(
,
' 
 y
x
y
1
)
0
(
,
1
' 

 y
y
xy
No solutions
Precisely one solutions
More than one solutions

Page 57
Existence and uniqueness theorems
 Problem of existence
 Under what conditions does an initial
value problem have at least one
solution ?
 Existence theorem, see page 53
 Problem of uniqueness
 Under what conditions does that the
problem have at most one solution ?
 Uniqueness theorem, see page54