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

王俊鑫 (Chun-Hsin Wang)

中華大學資訊工程系

Fall 2002

Chap 3 Linear Differential Equations

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

Outline

線性常微分方程線性

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

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

Second-Order ODE

General Form for Second-Order Linear ODE

Implicit Form

Explicit Form

0),,,( yyyxF

)()()( xryxqyxpy

Second-Order Homogeneous Linear Equations

Second-Order Homogeneous Linear ODE

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

Example

0)()( yxqyxpy

062)1( 2 yyxyx

Examples of Nonlinear differential equations

0'2)'( 2 yyyyyx

1'' 2 yy

A linear combination of Solutions for homogeneous linear equation

Example:

xx eyey ,

xx eey 53

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

Proof:

0)()()()(

))(())(()(

22221111

221122112211

yxqyxpycyxqyxpyc

ycycxqycycxpycyc

yxqyxpy

ycycyLet

yccy )(Note 2121 )( yyyy

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

Example: A nonhomogeneous linear differential equation

Example: A nonlinear differential equation

)sin1()cos1(

sin1,cos1

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

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

ececyyy xx

xx leyey 21 ,

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

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

sin,cos 21

How to obtain a Bass if One Solution is Known ?

Method of Reduction Order

Given y1

Find y2

0)()( yxqyxpy

Proof:

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

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

111111

1112112

uULetuy

yxqyxpyuyxpyuyu

uyxqyuyuxpyuyuyu

yxqyxpy

yuyuyuyyuyuy

uyyLet

Proof:

eyUdxyy

dxxpyU

)(ln2ln

Example 3-1:

212 ,,0 yFindxyyyxyx

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

Find Particular Solution

09 yy6)0( ,4)0( yy

Initial Condition

)3sin( ),3cos( xx

02 yyy0)0( ,1)0( yy

Initial Condition

-xx xee ,

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

Initial Condition

3/22/1 , xx

Exercise: Reduce of order if a solution is known.

2 ,09'5" xyyxyyx

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

02 baCharacteristicEquation

Case 1: 兩相異實根

Case 2: 重根

Case 3: 共軛虛根

042 ba

xx ececy 2121

xexccy 1)( 21

nim)sincos( nxBnxAey mx

Example 3-2:

Step 1: Find General Solution

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

xx ececy 221

Step 2: Find Particular Solution

Step 3: Plot Particular Solution

x=[0:0.01:2];

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

plot(x,y)

MATLAB Code

Case 2 Real Double Root = -a/2

2 ,04 axeyba

111111

1112112

0" ,0"

0'2,'2

0)()2(

0)()()2(

xexyyxutake

ayyayaey

byyayuayyuyu

uybyuyuayuyuyu

yuyuyuyyuyuy

uyyLet

Example 3-3:

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

xexccy 221

exccecy

x=[0:0.01:2];

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

plot(x,y)

MATLAB Code

Euler Formula

Proof:

xixeix sincos

!!3!21

!6!4!21)cos(

!7!5!3)sin(

xxxxxx

MaclaurinSeries

Euler Formula

Proof:

xixeix sincos

)sin()cos(

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

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

753642

765432

ixxixxixxix

ixixixixe

Euler Formula

1ie自然數

幾何

分析

虛數

負數

Complex Exponential Function

)sin(cos titeeeee

itszsitsitsz

Case 3 042 ba

)sincos(

sin,cos

)sin(cos

)2()2(

wxBwxAey

wxeywxey

wxiwxeee

xaiwxxax

Example 3-4:

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

)2sin2cos(

,001.42.0

xBxAey

)2cos22sin2(

)2sin2cos(1.0

)2sin2cos(

x=[0:0.1:30];

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

plot(x,y)

MATLAB Code

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

Modeling: Mass-Spring Systems

0 kyycym

Modeling: Electric Circuits

Inductor(heries)

Resistor(ohms)

Capacitor(farads)

Modeling 0 kyycym

kyycym

Modeling

Overdamping 042 mkc

tt ececty )(2

Modeling

Critical Damping 042 mkc

tetccty )()( 21

Modeling

Underdamping

042 mkc

twBtwAetyαt

),cos(

)sincos()(

Euler-Cauchy Equation

02 byyaxyx

)1(,, :122

bxamxxmm

bxaxmxxmmx

xmmymxyxySubstituteSol

0)1(2 bmamThe Auxiliary Equation

Case 1: Distinct Real Roots m1, m2

Example 3-5:

mm xcxcy

4 ,5.0

EquationAuxiliary :

00.25.22 yyxyx

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

mxxccy )ln( 21

Euler-Cauchy Case 2 :Example

Example 0432 yyxyx

EquationAuxiliary :

Case 3: Complex Roots m = a ± bi

)lnsin()lncos( xbBxbAxy a

Euler-Cauchy Case 3 :Example

Example 01372 yyxyx

)ln2sin()ln2cos(

EquationAuxiliary :

xBxAxy

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

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

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)

Wronskian

Example 3-8:

sincos

cossin

sincos)sin,(cos

xx sin ,cos

cosx, sinx are linearly independent

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 .

Illustration of Theorem 2

Example 1

Example 2

sinwxycoswxy

xexccy

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

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

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

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

Initial value problem for a nonhomogeneous equation

Example

9.0)0(',1.1)0(

4.10101'2"

eyyy x

Solution by Undetermined Coefficients

Method of Undetermined Coefficients

General Solution: y = yh + yp

yh : Homogeneous Solution

yp : Particular Solution

)(xrbyyay

xk cosxk sinxkenx cosxkenx sin

1 KxKxKxK nn

xKxK sincos 21 xKxK sincos 21

)sincos( 21 xKxKenx

Rules for the Method of Undetermined Coefficients

Basic Rule Modification Rule Sum Rule

Example 3-9:

8)42(44

xKKxKxK

xKxKxKK

KxKxKy

pyFindxyy ,84 2

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(

1.60)0(',2.0)0(

,4sin554cos4025.15'2 5.0

xxeyyy x

Second-Order Non-homogeneous Linear Equations

Method of Variation of Parameters

Particular Solution:

y1, y2 : Homogeneous Solutions

W : Wronskian of y1 and y2

)()()( xryxqyxpy

ryyxy p 1

Second-Order Non-homogeneous Linear Equations

Example 3-10:

2sin2cos22sin22sin

2cos2sin22cos22cos

2sin42sin22sin2cos42cos22cos

2cos82sin

2sin82cos

22cos22sin2

2sin2cos

2sin,2cos

xxxxxx

xdxxxxxxdxxxxx

pyFindxyy ,84 2

Higher Order Linear Differential Equations

Higher Order Homogeneous Linear ODE

0)()()( 01)1(

yxpyxpyxpy nn

If y1, y2, …, yn are the solutions of

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

0)()()( 01)1(

yxpyxpyxpy nn

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

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

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