‧
Textbook –
Advanced Engineering Mathematics, 9th Edition by Erwin Kreyszig,
ISBN 0-470-07446-9, 2006.
‧
Advanced Mathematics may include the following topics:
1. Ordinary Differential Equations (常微分方程式)
(ˇ)
2. Vector Analysis (向量分析) (ˇ)
3. Fourier Analysis (傅立葉分析) (ˇ)
4. Linear Algebra (線性代數) (３；選修)
5. Numerical Analysis (數值分析) (３；選修)
6. Complex Analysis (複變分析) (３)
7. Partial Differential Equations (偏微分方程式) (３)
-1-
PPaarrtt A
AO
Orrddiinnaarryy D
Diiffffeerreennttiiaall E
Eqquuaattiioonnss (常微分方程式：內含單變數函數的導數)
‧
Differential equations are of basic importance in engineering mathematics because many
physical laws and relations appear mathematically in the form of a differential equation.
Fig. 1 Applications of differential equations (微分方程式的應用)
-2-
C
Chhaapptteerr 11 FFiirrsstt--O
Orrddeerr O
OD
DE
Ess
Chapter 2 Second-Order Linear ODEs
Chapter 3 Higher-Order Linear ODEs
Chapter 5 Series Solutions of ODEs
Chapter 6 Laplace Transforms
11..11 B
Baassiicc C
Coonncceeppttss
‧ If you want to solve an engineering problem (usually of a physical nature), we first have
to formulate the problem as a mathematical expression in terms of variables, functions,
equations, and so forth. Such an expression is known as a mathematical model of the
given problem. (微分方程式是對應的工程問題的一種數學模型)
The process of setting up a model, solving it mathematically, and interpreting the result in
physical or other terms is called mathematical modeling (數學模型化) or, briefly, modeling (模型化).
‧ An ordinary differential equation (ODE) (常微分方程式) is an equation that contains
one or several derivatives of an unknown function, which we usually call y ( x ) or y ( t ) .
For example,
y  cos x ,
y   9 x  0 ,
x 2 yy  2e x y  ( x 2  2) y 2 .
The term ordinary distinguishes them from partial differential equations (PDEs) (偏微
分方程式), which involve partial derivatives of an unknown function of two or more variables (內含多變數函數的偏導數). For instance,
 2u
x 2
 yu2  0 .
2
PDEs are more complicated than ODEs. (偏微分方程式較常微分方程式來得複雜及難解)
-3-
‧ An ODE is said to be of order n (n 階) if the nth derivative of the unknown function y is
the highest derivative of y in the equation (方程式中所出現的最高階導數為 n 階).
The concept of order gives a useful classification into ODEs of first order, second order,
and so on. (通常，階數越高越難求解，故微分方程式常以其階數來分類)
‧
The most general form of first-order ODEs is
F ( x, y , y )  0 . (一階常微分方程式的通式)
Note that y  must be present, but x and/or y need not occur explicitly. (方程式中， y  的
成分必須存在，但 x 或 y 的成分未必存在)
If a first-order ODE can be expressed as
y   f ( x, y ) , (能表示成此種形式者叫做外顯式，否則叫做內隱式)
It is called the explicit form (外顯式), in contrast with the implicit form (內隱式).
‧
A solution (解) of F ( x, y, y )  0 on an interval I is a function h( x ) that satisfies the
equation for all x in I. (能讓方程式成立的函數稱之為解) That is,
F ( x, h( x ), h( x ))  0 for all x in I.
A solution contained an arbitrary constant is called the general solution (通解) of the differential equation. (包含一個任意常數的一階微分方程式的解，稱之為通解)
For example, h( x )  x ln x  cx is the general solution of y  
y
 1.
x
Each choice of the constant in the general solution yields a particular solution (特解).
For example, h( x )  x ln x  x is a particular solution of y  
y
 1.
x
‧ A solution is explicit (外顯式的解) if it is given as a function of independent variables.
For example, y  ke x is an explicit solution of y    y . Note that this general solution
-4-
is explicit, with y isolated on one side of an equation and a function of x on the other. (可
以單獨把 y 孤立出來的解稱之為外顯式的解)
By contrast, consider y   
2 xy 3  2
. The general solution is x 2 y 3  2 x  2e4 y 
2 2
4y
3x y  8e
k , which is implicit (內隱式的解). In this example we are unable to solve the differential
equation explicitly for y as a function of x while isolating y on one side. (不能單獨把 y 孤
立出來的解稱之為內隱式的解)
‧
The graph of a particular solution of a first-order differential equation is called an integral
curve (積分曲線) or a solution curve (解的曲線) of the equation.
Example 1
Verification of Solution (解的驗證)
Example 2
Solution Curves (解的曲線)
Example 3
Exponential Growth or Decay (指數型成長或衰減)
-5-
‧
If we specify that a particular solution is a solution passing through a particular point
( x 0 , y 0 ), then we have to find that particular integral curve passing through this point.
This is called an initial value problem (初始值問題).
‧
Thus, a first-order initial value problem has the form
F ( x, y , y )  0 ; y ( x0 )  y0 ,
in which x0 and y0 are given numbers. The condition y ( x0 )  y0 is called an initial
condition (初始條件).
Example 4
Initial Value Problem
‧ Homework for sec.1.1  #7, 9
-6-
11..22 D
Diirreeccttiioonn FFiieelldd ((方
方向
向場
場)) -- G
Geeoom
meettrriicc M
Meeaanniinngg ooff y  f ( x, y )
‧
Consider the general first-order differential equation of the form
F ( x, y , y )  0 .
Suppose we can solve for y  as
y   f ( x, y ) .
Then a drawing of the plane, with short line segments of slope f ( x, y ) drawn at selected
point (x,y), is called a direction field (方向場) or slope field (斜率場) of the differential
equation.
A direction field can be used to find approximate solutions, but with limited accuracy. (方
向場可用於近似解之求取，儘管準確度較為有限)
‧
A famous ODE for which we do need direction field is
y   0.1(1  x 2 ) 
x
.
y
The direction field in Fig. 8 shows lineal elements generated by the computer. We have
also added the isoclines (等斜率線) for k  5 , 3 ,
1
4
, 1 as well as three typical solu-
tion curves, one that is a circle and two spirals approaching it from inside and outside.
‧ Homework for sec.1.2  #1, 5, 13
-7-
11..33 SSeeppaarraabbllee O
OD
DE
Ess ((可
可分
分離
離型
型微
微分
分方
方程
程式
式)) g ( y ) y  f ( x )
‧
Many practically useful ODEs can be reduced to the separable form
g ( y ) y  f ( x )
or
g ( y )dy  f ( x )dx .
The variables are separated: x appears only on the right and y only on the left. And we can
integrate the differential equation along any solution curve to get
 g ( y )dy   f ( x)dx .
In case f(x) and g(y) are continuous functions, we can obtain a general solution from the
above integrals.
Example 1
A separable ODE (可分離型微分方程式)
Example 2
Radiocarbon Dating (放射性碳定年)
-8-
Example 3
‧
Mixing Problem (混合問題)
Certain non-separable ODEs can be made separable by transformations that introduce for
y a new unknown function. (某些非可分離型微分方程式，可透過新函數的引進，將之
轉換成可分離型)
For a homogeneous ODE, say
y   f ( xy ) ,
we can set u 
y
x
to get y  ux and y   ux  u . Substituting them into the homoge-
neous ODE, we have
-9-
ux  u  f (u )

du
dx
,

f (u )  u x
which is a separable differential equation.
Example 6
Reduction to Separable Form (轉換為可分離型)
‧ Homework for sec.1.3  #5, 9, 11, 13, 33
- 10 -
11..44 E
Exxaacctt O
OD
DE
Ess ((正
正合
合型
型微
微分
分方
方程
程式
式)) M ( x, y )  N ( x, y ) y   0
‧
If a function u ( x, y ) has continuous partial derivatives, its total differential (全微分) is
du 
u
u
dx  dy .
x
y
It follows that along any contour lines (等高線) of u( x, y )  c  constant, we have
du 
‧
u
u
dx  dy  0 .
x
y
A first-order ODE written as
M ( x, y )  N ( x, y ) y   0
or
M ( x, y )dx  N ( x, y )dy  0 .
is called exact (正合的) if there exists a function u ( x, y ) such that
M ( x, y ) 
And we have
‧
u
x
u
u
and N ( x, y ) 
.
x
y
dx  uy dy  0 .
By integration, we immediate obtain the general solution of
u
x
dx  uy dy  0 as
u ( x, y )  c .
This is called an implicit solution (內隱解), in contrast with an explicit solution (外顯解)
of the form y  y ( x ) .
‧
Let M ( x, y ) and N ( x, y ) have continuous first partial derivatives. Then we have
M
 2u
,

y yx
N
 2u
.

x xy
By the assumption of continuity, the two second partial derivatives are equal. Thus we
have the following necessary and sufficient condition (充分必要條件)
M ( x, y ) N ( x, y )
. (Test for exactness)

y
x
- 11 -
‧
If M ( x, y )  N ( x, y ) y   0 is exact, that is M ( x, y ) 
u
x
and N ( x, y ) 
function u ( x, y ) can be found in the following systematic way:
1) find u( x, y )   ux dx   M ( x, y )dx ;
2) find u( x, y )   uy dy   N ( x, y )dy ;
3) compare the results of 1) and 2) to determine u ( x, y ) ;
4) the general solution is u( x, y )  c  const.
Example 1
Solve
An Exact ODE (正合型微分方程式之求解)
cos( x  y )dx  (3 y 2  2 y  cos( x  y ))dy  0 .
- 12 -
u
y
, then the
‧ An integrating factor (積分因子) is such a function that a multiplication of this factor
with the differential equation will result in an equation that can be integrated to obtain the
general solution.
For the inexact (非正合) differential equation M ( x, y )  N ( x, y ) y   0 , the general integrating factor is F ( x, y ) provided ( FM )  ( FN ) y  0 is exact or

y
( FM ) 

x
( FN ) .
In case F  F ( x ) , we have

y
( FM ) 

x
( FN )

1 dF ( x )
F ( x ) dx

1
N ( x, y )
 F ( x)  exp N ( x , y )
1
[ M(yx, y )  N ( xx, y ) ]
[
M ( x , y ) N ( x , y )
 x ]dx .
y
In case F  F ( y ) , we have

y
( FM ) 

x
( FN )

1 dF ( y )
F ( y ) dy

1
M ( x, y )
 F ( y )  exp M ( x , y )
1
[ N ( xx , y )  M (yx, y ) ]
[
N ( x , y ) M ( x , y )
 y ]dy .
x
‧ Homework for sec.1.4  #1, 5, 13, 17, 21, 24(D)
11..55 L
Liinneeaarr O
OD
DE
Ess ((線
線性
性微
微分
分方
方程
程式
式)) y  p( x ) y  r ( x )
‧
A first-order ODE is said to be linear (線性的) if it can be written as
y  p( x ) y  r ( x ) .
(1)
In engineering, r ( x ) is frequently called the input (輸入), and y ( x ) is called the output (輸出) or the response (反應;響應) to the input.
‧
For the linear differential equation y  p( x ) y  r ( x ) , the general integrating factor is
- 13 -
e
p ( x ) dx
. Thus multiply eq.(1) on both sides by e 
e
p ( x ) dx

 ( y   p( x ) y )  e 
p ( x ) dx
p ( x ) dx
to get
 r( x) .
 e  p ( x ) dx  y   e p ( x ) dx  r( x)




Integrate this equation with respect to x to get
Example 1
e

 p ( x dx
) 
p
ye 
e

p ( x ) dx
y
p ( x ) dx
r ( x )dx
r ( x )dx 

x( dx)
A Linear ODE (線性微分方程式之求解)
Solve the linear ODE
Example 2
  e

y  y  e2 x .
Initial Value Problem (初始值問題)
Solve the initial value problem
y  y tan x  sin 2 x ,
- 14 -
y (0)  1 .
‧
Numerous applications can be modeled by ODEs that are nonlinear but can be transformed to linear ODEs. One of the most useful ones of these is the Bernoulli equation
y   p( x ) y  r ( x ) y a ,
a is any real number.
(2)
In case a  0 or a  1 , equation (2) is linear. Otherwise, it is nonlinear.
To solve it, let a new function
u( x )  [ y ( x)]1a .
Differentiate it on both sides to get
u  (1  a ) y  a y
 (1  a ) y  a ( ry a  py )
 (1  a )( r  py1a )
 (1  a )( r  pu )
 u  (1  a ) pu  (1  a )r , which is linear.
Example 4
Logistic Equation (數理邏輯方程式)
Solve the Bernoulli equation, known as the logistic equation
y  A y B2 y.
- 15 -
(3)
The logistic equation (3) plays an important role in population dynamics (人口動態學), a
field that models the evolution of populations of plants, animals, or humans over time t. In
case B  0 , the solution gives exponential growth as for a small population in a large country
(the United States in early times!). The term  By 2 in equation (3) is a braking term (抑制項)
that prevents the population from growing without bound.
‧ Homework for sec.1.5  #3, 7, 11, 15, 18, 22, 27
11..66 O
Orrtthhooggoonnaall T
Trraajjeeccttoorriieess ((正
正交
交軌
軌跡
跡))
‧
An important type of problem in physics or geometry is to find a family of curves that intersect a given family of curves at right angles. These new curves are called orthogonal
trajectories (正交軌跡) of the given curves (and conversely). Orthogonal (正交) is another word for perpendicular (垂直).
- 16 -
‧
In many cases, orthogonal trajectories can be found by using ODEs as follows: Let
G ( x, y , c )  0
be a given family of curves in the xy-plane, where each curve is specified by some value
of c. This is called a one-parameter family of curves, and c is called the parameter of the
family. For instance,
G( x, y, c)  y  cx 2  0 . (parabolas;雙曲線)
Step 1
Find an ODE for which the given family is a general solution.
In our example, we solve algebraically for c from the equation of family of curves,
y
 c,
x2
and then differentiate and simplify to get
x 2 y  2 xy
 0,
x4
hence
y 
2y
.
x
It is the ODE of the given family of curves.
Step 2
Write down the ODE of the orthogonal trajectories.
This ODE is
y 
Step 3
1
1
x
  2y  
.
y
2y
x
Solve the ODE.
The equation y    2xy is separable. It can be rewritten as 2 ydy   xdx . By integration,
we have the orthogonal trajectories of the parabolas y  cx 2  0 as
y 2  12 x 2  c* , c*  0 . (ellipses;橢圓)
- 17 -
‧ Homework for sec.1.6  #3, 9, 16, 18
SSU
UM
MM
MA
AR
RY
Y
‧ 一階常微分方程式有標準解法的三種基本類型：
No.
Type/類型
Standard form/標準式
1
separable
g ( y ) y  f ( x )
Algorithm/求解步驟
1.變數分離 g ( y )dy  f ( x )dx
 g ( y )dy   f ( x)dx
2.積分
1.若為 exact, 則存在一 potential function u( x, y )
M ( x, y )  N ( x, y ) y   0
2
Exact
with
M
y

N
x
2.令 ux  M ( x, y ) , 求解 u( x, y )  M ( x, y )dx

3.令 uy  N ( x, y ) , 求解 u( x, y )  N ( x, y )dy

4.比較 2 與 3 之結果得 u( x, y ) ，則通解為 u( x, y )  const
1.存在一個積分因子 e  p ( x ) dx
2.微分方程式兩邊乘以積分因子 e  p ( x ) dx ，
3
linear
y  p( x ) y  r ( x )
得
d
dx
e
 p ( x ) dx
 y   e  p ( x ) dx  r ( x )
3.兩邊對 x 積分得 e  p ( x ) dx  y  e  p ( x ) dx r ( x )dx

‧ 一階常微分方程式之特殊類型：
No.
1
2
Type/類型
homogeneous
Bernoulli
Standard form/標準式
1.利用變數變換法，令 u  xy ，代回原式
y   f ( xy )
y   p( x ) y  r ( x ) y
Algorithm/求解步驟
2.將原式轉換成 separable 型
a
1.利用變數變換法，令 u  y1a ，代回原式
2.將原式轉換成 linear 型
- 18 -