王俊鑫 (Chun-Hsin Wang) 中華大學資訊工程系 Fall 2002 Chap 3 Linear Differential Equations

王俊鑫 (Chun-Hsin Wang)

中華大學資訊工程系

Fall 2002

Chap 3 Linear Differential Equations

Chap 3 Linear Differential Equations

Page 2

Outline

Second-Order Homogeneous Linear Equations Second-Order Homogeneous Equations with

Constant Coefficients Modeling: Mass-Spring Systems, Electric Circuits Euler-Cauchy Equation Wronskian Second-Order Nonhomogeneous Linear

Equations Higher Order Linear Differential Equations

Page 3

Outline

線性常微分方程線性

常微分方程二階

常微分方程二階

常微分方程

高階線性常微分方程高階線性常微分方程

二階線性常微分方程二階線性常微分方程

二階線性齊次常微分方程二階線性齊次常微分方程

二階線性非齊次常微分方程

二階線性非齊次常微分方程

常係數二階線性齊次常微分方程

常係數二階線性齊次常微分方程

歐拉 -柯西微分方程歐拉 -柯西微分方程

Page 4

Second-Order ODE

General Form for Second-Order Linear ODE

Implicit Form

Explicit Form

0),,,( yyyxF

)()()( xryxqyxpy

Page 5

Second-Order Homogeneous Linear Equations

Second-Order Homogeneous Linear ODE

p(x), q(x): coefficient functions

Example

0)()( yxqyxpy

062)1( 2 yyxyx

Page 6

Examples of Nonlinear differential equations

0'2)'( 2 yyyyyx

1'' 2 yy

Page 7

A linear combination of Solutions for homogeneous linear equation

Example:

xx eyey ,

0" yy

xx eey 53

Page 8

Linear Principle (Superposition Principle)

y is called the linear combination of y1 and y2

If y1 and y2 are the solutions of

y = c1y1+ c2y2 is also a solution(c1, c2 arbitrary constants)


0)()( yxqyxpy

Page 9


Proof:

0)()()()(

))(())(()(

)()(

22221111

221122112211

2211

yxqyxpycyxqyxpyc

ycycxqycycxpycyc

yxqyxpy

ycycyLet

yccy )(Note 2121 )( yyyy

Page 10

Does the Linearity Principle hold for nonhomogeneous linear or nonlinear equations ?

Example: A nonhomogeneous linear differential equation

Example: A nonlinear differential equation

)sin1()cos1(

1"

sin1,cos1

xx

yy

xyxy

1

0'"

1,

2

2

x

xyyy

yxy

Page 11

Initial Value Problem for Second-Order homogeneous linear equations

For second-order homogeneous linear equations,

a general solution will be of the form

, a linear combination of two solutions involving two arbitrary constants c1 and c2

An initial value problem consists two initial conditions.

0)()( yxqyxpy

1000 )(',)( kxykxy

2211 ycycy

Page 12

Initial Value Problem

Example:

Observation:

Our solution would not have been general enough to satisfy the two initial conditions and solve the problem.

2)0(',4)0(

,0" 21

yy

ececyyy xx

xx leyey 21 ,

Page 13

A General Solution of an Homogeneous Linear Equation

Definition: A general solution of an equation

on an open interval I is a solution

with y1 and y2 not proportional solutions of the equation on I and c1 ,c2 arbitrary constants.

The y1 and y2 are then called a basis (or fundamental system) of the equation on I

A particular solution of the equation is obtained if we assign specific values to c1 ,c2

0)()( yxqyxpy

2211 ycycy

Page 14

Linear Independent

Two functions y1(x) and y2(x) are linear independent on an interval I where they are defined if

Example

0,00)()( 212211 kkxykxyk

0"

sin,cos 21

yy

xyxy

Page 15

How to obtain a Bass if One Solution is Known ?

Method of Reduction Order

Given y1

Find y2

0)()( yxqyxpy

dxy

eyy

dxxp

21

)(

12

Page 16


Proof:

0)(2

,0)(2

0))()(())(2(

0))(())(()2(

0)()(

2,

1

11

1

11

111111

111111

1112112

12

Uy

yxpyU

uULetuy

yxpyu

yxqyxpyuyxpyuyu

uyxqyuyuxpyuyuyu

yxqyxpy

yuyuyuyyuyuy

uyyLet

Page 17


Proof:

dxy

eyUdxyy

y

eU

dxxpyU

dxy

yxpy

U

dU

dxxp

dxxp

21

)(

112

21

)(

1

1

11

)(ln2ln

)(2

Page 18


Example 3-1:

Sol:

212 ,,0 yFindxyyyxyx

xx

dxx

xdxx

ex

dxx

exdx

y

eyy

yx

yx

y

x

dxx

dxxp

ln

1

011

2

ln

2

1

21

)(

12

2

Page 19

Second-Order Homogeneous Linear Equations Exercise 3-1: Basic Verification and

Find Particular Solution

09 yy6)0( ,4)0( yy

Basis

Initial Condition

)3sin( ),3cos( xx

02 yyy0)0( ,1)0( yy

Basis

Initial Condition

-xx xee ,

034 2 yyx5.2)1( ,3)1( yy

Basis

Initial Condition

3/22/1 , xx

Page 20

Exercise: Reduce of order if a solution is known.

31

2 ,09'5" xyyxyyx

Page 21

Second-Order Homogeneous Equations with Constant Coefficients

General Form of Second-Order

Homogeneous Equations with Constant

Coefficients

whose coefficients a and b are constant.

0 byyay

Page 22


Sol:

0)(

0

0)()(

0

2

2

x

xxx

xxx

x

eba

beeae

beeae

eyTry

byyay

02 baCharacteristicEquation

Page 23

Case 1: 兩相異實根

Case 2: 重根

Case 3: 共軛虛根


042 ba

042 ba

042 ba

xx ececy 2121

xexccy 1)( 21

21,

11,

nim)sincos( nxBnxAey mx

Page 24


Example 3-2:

Sol:

Step 1: Find General Solution

5)0(,4)0(,02 yyyyy

xx ececy 221

2

2,1

,02

Page 25


Step 2: Find Particular Solution

xx

xx

xx

eey

cc

ccy

ccy

ececy

ececy

2

21

21

21

221

221

3

3,1

52)0(

4)0(

2

Page 26


Step 3: Plot Particular Solution

x=[0:0.01:2];

y=exp(x)+3*exp(-2*x);

plot(x,y)

MATLAB Code

Page 27

Case 2 Real Double Root = -a/2

21

2 ,04 axeyba

221

212

21

1

1112

1

111

111111

111111

1112112

12

)(

,

0" ,0"

0'2,'2

0

0)()2(

0)()()2(

0

2,

ax

ax

ax

exccy

xexyyxutake

cxcu

uyu

ayyayaey

byyay

byyayuayyuyu

uybyuyuayuyuyu

byyay

yuyuyuyyuyuy

uyyLet

Page 28


Example 3-3:

Sol:


1)0(,3)0(,044 yyyyy

xexccy 221

2

)(

2,2

,044

Page 29



x

xx

x

exy

cc

ccy

cy

exccecy

exccy

2

21

12

1

221

22

221

)53(

5,3

12)0(

3)0(

)(2

)(

Page 30



x=[0:0.01:2];

y=(3-5*x).*exp(2*x);

plot(x,y)

MATLAB Code

Page 31

Euler Formula

Euler Formula

Proof:

xixeix sincos

0

32

!!3!21

n

nx

n

xxxxe

0

2642

)!2(

)1(

!6!4!21)cos(

n

nn

n

xxxxx

0

12753

)!12(

)1(

!7!5!3)sin(

n

nn

n

xxxxxx

MaclaurinSeries

Page 32

Euler Formula

Proof:

xixeix sincos

)sin()cos(

!7!5!3!6!4!21

!7!6!5!4!3!21

!

)(

!3

)(

!2

)(1

753642

765432

0

32

xix

xxxxi

xxx

ixxixxixxix

n

ixixixixe

n

nix

Page 33

Euler Formula

1ie自然數

幾何

分析

虛數

負數

Page 34

Complex Exponential Function

)sin(cos titeeeee

itszsitsitsz

Page 35

Case 3 042 ba

)sincos(

sin,cos

)sin(cos

)sin(cos

4

1

,2

,2

2

22

21

)2()2(

)2()2(

2

21

2

1

wxBwxAey

wxeywxey

wxiwxeee

wxiwxeee

abw

iwa

iwa

ax

axax

xaiwxxax

xaiwxxax

Page 36


Example 3-4:

Sol:


2)0(,0)0(,001.42.0 yyyyy

)2sin2cos(

21.0

,001.42.0

1.0

2

xBxAey

ix

Page 37



xey

BA

By

Ay

xBxAe

xBxAey

xBxAey

x

x

x

x

2sin

1,0

22)0(

0)0(

)2cos22sin2(

)2sin2cos(1.0

)2sin2cos(

1.0

1.0

1.0

1.0

Page 38



x=[0:0.1:30];

y=exp(-0.1*x).*sin(2*x);

plot(x,y)

MATLAB Code

Page 39


Exercise 3-2: Find General Solution

0344 yyy

092 yy

044 yyy

025309 yyy

022 yyy

兩相異實根

01044 yyy

重根

共軛虛根

Page 40

Modeling: Mass-Spring Systems

0 kyycym

Page 41

Modeling: Electric Circuits

Inductor(heries)

Resistor(ohms)

Capacitor(farads)

01

IC

IRIL

Page 42

Modeling 0 kyycym

21

2

22,1

2

,

42

1,

2

42

1

2

0

0

mkcmm

c

mkcmm

cm

k

m

c

kyycym

Page 43

Modeling

Overdamping 042 mkc

tt ececty )(2

)(1)(

Page 44

Modeling

Critical Damping 042 mkc

tetccty )()( 21

Page 45

Modeling

Underdamping

042 mkc

ABBAC

twCe

twBtwAetyαt

t

tan,

),cos(

)sincos()(

22

*

**

Page 46

Euler-Cauchy Equation


02 byyaxyx

0)1(

0)1(

)1(,, :122

21

mmm

mmm

mmm

bxamxxmm

bxaxmxxmmx

xmmymxyxySubstituteSol

0)1(2 bmamThe Auxiliary Equation

Page 47


Case 1: Distinct Real Roots m1, m2

Example 3-5:

2121

mm xcxcy

42

5.01

2

4 ,5.0

025.3

EquationAuxiliary :

xcxcy

m

mm

Sol

00.25.22 yyxyx

Page 48


Case 2: Double Roots m=(1-a)/2

mxxccy )ln( 21

Page 49

Euler-Cauchy Case 2 :Example

Example 0432 yyxyx

221

2

)ln(

2 ,2

044

EquationAuxiliary :

xxccy

m

mm

Sol

Page 50


Case 3: Complex Roots m = a ± bi

)lnsin()lncos( xbBxbAxy a

Page 51

Euler-Cauchy Case 3 :Example

Example 01372 yyxyx

)ln2sin()ln2cos(

23

0136

EquationAuxiliary :

3

2

xBxAxy

im

mm

Sol

Page 52

Existence and Uniqueness Theory

If p(x) and q(x) are continuous function on some open interval and x0 is in , then the initial value problem consisting of (1) and (3) has a unique solution y(x) on the interval .

)1(0)()( yxqyxpy

)3()(',)( 1000 kxykxy

)2(2211 ycycy

Page 53

Wronskian

A set of n functions y1(x), y2(x), …, yn(x),

is said to be linearly dependent over an interval I if there exist n constants c1, c2, …, cn, not all zero, such that

Otherwise the set of functions is said to be linearly independent

0)()()( 2211 xycxycxyc nn

Page 54

Wronskian

A set of n functions y1(x), y2(x), …, yn(x), is

linearly independent over an interval I if and only if the determinant (Wronski determinant, or Wronskian)

0

),,,(

)()(2

)(1

21

21

21

nn

nn

n

n

n

yyy

yyy

yyy

yyyW

Page 55

Wronskian

Example 3-8:

Sol:

01

sincos

cossin

sincos)sin,(cos

22

xx

xx

xxxxW

xx sin ,cos

cosx, sinx are linearly independent

Page 56

Linear Dependence and Independence of Solution

Suppose that (1) has continuous coefficients p(x) and q(x) on an open interval . Then two solutions y1 and y2 of (1) on are linear dependent on if and only if their Wronskian W is zero at some x0 in .

Furthermore, if W=0 for x= x0, then W=0 on ; hence if there is an x1in at which W is not zero, then y1 ,y2 are liner independent on .

Page 57

Illustration of Theorem 2

Example 1

Example 2

0"

,2

21

ywy

sinwxycoswxy

xexccy

yyy

)(

,0'2"

21

Page 58

A General Solution of (1) includes All Solutions

Theorem 3 (Existence of a general solution)If p(x) and q(x) are continuous on an open interval , then (1) has a general solution on .

Theorem 4 (General solution)Suppose that (1) has continuous coefficients p(x) and q(x) on some open interval . Then every solution y=Y(x) of (1) is of the form

where y1 , y2 form a basis of solutions of (1) on and c1, c2 are

suitable constants. Hence (1) does not have singular solutions (I.e., solutions not obtainable from a general solution)

)1(0)()( yxqyxpy

)()( 2211 xycxycY

Page 59

Nonhomogeneous Equations

Theorem (a) The difference of two solutions of (1)

on some open interval is a solution of (2) on

(b) The sum of a solution of (1) and a solution of (2) on is a solution of (1) on

)2(0)()(

)1()()()(

yxqyxpy

xryxqyxpy

Page 60

A general solution of the nonhomogeneous equation (1) on some open interval is a solution of the form

where yh(x)=c1y1(x)+c2y2(x) is a general solution of the homogeneous equation (2) on and yp(x) is any solution of (1) on containing no arbitrary constants.

A particular solution of (1) on is a solution obtain from (3) by assigning specific values to the arbitrary constants c1 and c2 in yh(x).

)2(0)()(

)1()()()(

yxqyxpy

xryxqyxpy

)3()()()( xyxyxy ph

Page 61

Practical Conclusion

To solve the nonohomegeneous equation (1) or an initial value problem for (1) , we have to solve the homogeneous equation (2) and find any particular solution yp of (1)

)2(0)()(

)1()()()(

yxqyxpy

xryxqyxpy

Page 62

Initial value problem for a nonhomogeneous equation

Example

9.0)0(',1.1)0(

4.10101'2"

yy

eyyy x

Page 63

Solution by Undetermined Coefficients

Method of Undetermined Coefficients

General Solution: y = yh + yp

yh : Homogeneous Solution

yp : Particular Solution

)(xrbyyay

Page 64


nkx

nxke

)(xr

xk cosxk sinxkenx cosxkenx sin

py

011

1 KxKxKxK nn

nn

xKxK sincos 21 xKxK sincos 21

)sincos( 21 xKxKenx

)sincos( 21 xKxKenx

nxke

Page 65

Rules for the Method of Undetermined Coefficients

Basic Rule Modification Rule Sum Rule

Page 66


Example 3-9:

Sol:

12

1,0,2

8)42(44

8)(42

2

2

012

2021

22

201

222

2

012

2

xy

KKK

xKKxKxK

xKxKxKK

Ky

KxKxKy

p

p

p

pyFindxyy ,84 2

Page 67

Example for Modification Rule

Example 1: in the case of a simple root

Example 2: in the case of a double root

Example 3: sum rule.

px yFindeyyy ,2'3

1)0(',1)0(

,1'2

yy

eyy x

1.60)0(',2.0)0(

,4sin554cos4025.15'2 5.0

yy

xxeyyy x

Page 68

Second-Order Non-homogeneous Linear Equations

Method of Variation of Parameters

Particular Solution:

y1, y2 : Homogeneous Solutions

W : Wronskian of y1 and y2

)()()( xryxqyxpy

dxW

ryydx

W

ryyxy p 1

22

1)(

Page 69

Second-Order Non-homogeneous Linear Equations

Example 3-10:

Sol:

12

2sin2cos22sin22sin

2cos2sin22cos22cos

2sin42sin22sin2cos42cos22cos

2

2cos82sin

2

2sin82cos

22cos22sin2

2sin2cos

2sin,2cos

2

2

2

22

22

12

21

21

x

xxxxxx

xxxxxx

xdxxxxxxdxxxxx

dxxx

xdxxx

x

dxW

ryydx

W

ryyy

xx

xxW

xyxy

p

pyFindxyy ,84 2

Page 70

Higher Order Linear Differential Equations

Higher Order Homogeneous Linear ODE

0)()()( 01)1(

1)(

yxpyxpyxpy nn

n

If y1, y2, …, yn are the solutions of

y = c1y1+ c2y2 +… + cnyn will be the general solution

0)()()( 01)1(

1)(

yxpyxpyxpy nn

n

Page 71

Higher Order Linear Differential Equations

Higher Order Nonhomogeneous Linear ODE

General Solution: y = yh + yp

yh : Homogeneous Solution

yp : Particular Solution

)()()()( 01)1(

1)( xryxpyxpyxpy n

nn

Documents

王 俊 鑫 (Chun-Hsin Wang) 中華大學 資訊工程系 Fall 2002 Chap 3 Linear Differential Equations

王俊鑫 (Chun-Hsin Wang) 中華大學資訊工程系 Fall 2002 Chap 3 Linear Differential Equations