Chapter 4: Higher-Order Differential Equations

Chapter 4: Higher-Order Differential Equations

Math-202

CH#1

Difinitions

CH#2

1st order DE

CH#4

Higher order DE

Chapter 4: Higher-Order Differential Equations

xyyexyy x cos7'''5'''3

xyyxyy cos7'''5'''3 2

xyyxyy coscos7'''5'''3

xyyxyy cos7'''5'''3

Sec 4.1: Linear DE (Basic Theory)

Sec 4.1.1: Initial Value Problem (IVP)

Boundary Value Problem (BVP)

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

1)( xgyxayxayxayxa n

nn

n

,)( ,)(' ,)( 10)1(

1000 n

n yxyyxyyxy IVP: . nth order linear DE

Theroem 4.1 ( Existence of a Unique Solution)

Ixa

Ixgxaxaxaxa

Ix

n

nn

on 0)( )3

on cont are )( ),( ),(,),( ),( )2

)1

011

0

Sol y(x)

Exist

unique

:Example 0)1('' ,0)1(' ,0)1( ,cos7'''5'''3 yyyxyyexyy x1

Sec 4.1: Linear DE (Basic Theory)

Theroem 4.1 ( Existence of a Unique Solution)

Ixa

Ixgxaxaxaxa

Ix

n

nn

on 0)( )3

on cont are )( ),( ),(,),( ),( )2

)1

011

0

Sol y(x)

Exist

unique

:Example 0)1('' ,0)1(' ,0)1( ,cos7'''5'''3 yyyxyyexyy x1

10 40 124 )y'(,)y(x,yy''2 xeexy xx 33)( 22

10 30 62'22 )y'(,)y(,yxyy''x3 3)( 2 xxxy37)( 2 xxxy

Sec 4.1: Linear DE (Basic Theory)

Theroem 4.1 ( Existence of a Unique Solution)

Ixa

Ixgxaxaxaxa

Ix

n

nn

on 0)( )3

on cont are )( ),( ),(,),( ),( )2

)1

011

0

Sol y(x)

Exist

unique

:Example

10 00 3'')2( )y'(,)y(x,yyx9/p138

Find an interval centered about x=0 for which the given IVP has a unique solution

10 00 10)(tan'4

3'')25( 2

)y'(,)y(,xyxy

xyx2

Sec 4.1: Linear DE (Basic Theory)

50 40 124 )y'(,)y(x,yy''

Problem 1

51 40 124 )y(,)y(x,yy''

Problem 2

2ed order linear DE

What is the difference

IVP BVP

Sec 4.1: Linear DE (Basic Theory)

)()()()(

0

012

,yy(a)

,xgyxay'xay''xa

2ed order linear DE

IVP BVP

)()()()(

0

012

,yy(a)

,xgyxay'xay''xa

1' y(a)y 1yy(b)

Sec 4.1: Linear DE (Basic Theory)

10

012

)()()()(

yy'(a),yy(a)

,xgyxay'xay''xa

2ed order linear DE

IVP BVP10

012

)()()()(

yy(b),yy(a)

,xgyxay'xay''xa

Exist and unique

When??

BVP can have many, one, or No sol

0)8

(

0)0(

016''

y

y

yy

:Example

1)2

(

0)0(

016''

y

y

yy

0)2

(

0)0(

016''

y

y

yy

BVP3BVP2BVP1

xcxcxy 4sin4cos)( 21 2-parameter family of solutions Given that

0)(

0

0

2

1

xy

c

c

solutionno

c

01

01

xcxy

c

4sin)(

01

uniqueNo sol Infinity number of sol

Sec 4.1.2: Homogeneous Equations

0)(')()()( 01)1(

1)(

yxayxayxayxa nn

nn

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

1)( xgyxayxayxayxa n

nn

n

diff

homogeneous nonhomogeneous

:Example

(*) 0cos7'''5'''3 xyyexyy x1

2 (**) 07'''5'''3 yyexyy x

(**) is the associated homogeneous DE of (*)

Remark: before we solve (*), we have to solve first (**)

Differential Operator

dx

dD

)( )1 2xD

:Example

)sin( )2 3 xxD

)( )3 22 xD

)3)(252( )4 2423 xxDDD

Differential Operators

operator polynomial

operator aldifferentiorder th -n

)()()()( 011

1 xaDxaDxaDxaL nn

nn

)3)(25( )5 24 xxxD

Properties: Differential Operator

aL[f(x)]L[af(x)] )1

L[g(x)]L[f(x)]g(x)]L[f(x) )2

)()()()( 011

1 xaDxaDxaDxaL nn

nn

bL[g(x)]aL[f(x)]bg(x)] L[af(x) )3

Linear Operator

Quiz on Monday

2.1

3.1

4.1.1

DE Differential Operator Form

:Example

356'5'' xyyy

xyyxxy sin6'5''' 2

gyL )(

gyL )(

:Example

33 xDDL xxg sinwhere

Write as DE

Homog DE

Theroem 4.2 ( Superposition Principle)

(*) of solutions twobe and Let 21 yy operatorlinear a is where L(*) 0L(y)sol also is c 1y

sol also is y21 y

:Example0423 yxy'y'''-x

21 xy

xxy ln22

are solutions1)Constant multiple is sol

2)Sum of two sol is also sol

3) Trivial sol is also a sol ??

Homog DE

Theroem 4.2 ( Superposition Principle)

(*) of solutions twobe ,,y,y,Let 321 kyy operatorlinear a is where L(*) 0L(y)

sol also is 2211 kk ycycyc

In general

Linear Dependence & Linear Independence

Definition 4.1

I(x), f(x),(x),ff n on dependent linearly functions 21 IF

such that zeros allnot , constantsexit there 21 nc,,cc

02211 (x) fc(x)fc(x)fc nn

(x) fc(x)fc(x)fc nn 2211Note: Linear Combination

:Example

32 21 x(x)f

122 x(x)f

32 (x)f 1

,6

,3

3

2

1

c

c

cIs this set linearly dependent ??

IF not then we say linearly independent

for every x in I

Linear Dependence & Linear Independence

Definition 4.1

I(x), f(x),(x),ff n on dependent linearly functions 21 IF

such that zeros allnot , constantsexit there 21 nc,,cc

02211 (x) fc(x)fc(x)fc nn

:Example

x(x)f 21 cos

x(x)f 22 sin

12 (x)f 1

,1

,1

3

2

1

c

c

cIs this set linearly dependent ??

IF not then we say linearly independent

for every x in I

Linear Dependence & Linear Independence

Definition 4.1

I(x), f(x),(x),ff n on dependent linearly functions 21 IF

such that zeros allnot , constantsexit there 21 nc,,cc

02211 (x) fc(x)fc(x)fc nnIF not then we say linearly independent

Special case dep. lin. are and 21 (x)f(x)fIf a set of two functions is lin. Dep, then one function is simply a constant multiple of the other.

:Examplex(x)f 2sin1

xx(x)f cossin2

Is this set linearly dependent ??

for every x in I

Linear Dependence & Linear Independence

Definition 4.1

I(x), f(x),(x),ff n on dependent linearly functions 21 IF

such that zeros allnot , constantsexit there 21 nc,,cc

02211 (x) fc(x)fc(x)fc nnIF not then we say linearly independent

:Examplex(x)f 1

x(x)f 2

Is this set linearly dependent ??

for every x in I

Linear Dependence & Linear Independence

Definition 4.1

I(x), f(x),(x),ff n on dependent linearly functions 21 IF

such that zeros allnot , constantsexit there 21 nc,,cc

02211 (x) fc(x)fc(x)fc nnIF not then we say linearly independent

Remark

A set of functions is linearly dependent if at least one function can be expressed as a linear combination of the remaining

:Example2

1 x(x)f x(x)f 2

Is this set linearly dependent ??

for every x in I

13 (x)f

532 24 xx(x)f

Homogeneous Equations

0)(')()()( 01)1(

1)(

yxayxayxayxa nn

nn

homogeneous

We are interested to find n linearly independent solutions

of the homog DE

)(,),(),(),( 321 xyxyxyxy n

Wronskian

Definition 4.2tdeterminan The s.derivative 1-n as Suppose 21 h(x), f(x),(x),ff n

called the Wronskian of the functions

)1()1(2

)1(1

21

21

21

'''),,,(

nn

nn

n

n

n

fff

fff

fff

fffW

:Example2

1 x(x)f x(x)f 2

Compute the Wroskian of these functions

23 x(x)f x(x)f 2

Compute the Wroskian of these functions

11 (x)f

Criterion for Linearly Independent Solutions

Theroem 4.3

:Examplexe(x)y 3

1 xe(x)y 32

These functions are solutions for the DE

0)(')()()( 01)1(

1)(

yxayxayxayxa nn

nn

THEN I,on DE homog for the solutions-n ,,, 21 nyyy

nyyy ,,, 21 Linearly Independent IxyyyW n in every for 0),,,( 21

09'' yy

lin. Indep ?

Fundamental set of solutions

:Example xe(x)y 31 xe(x)y 3

2

These functions are solutions for the DE 0'9''' yy

Fund. Set of sol. ?

Def 4.3 0)(')()()( 01)1(

1)(

yxayxayxayxa nn

nn

n are they ,,, )1 21 nyyy solutions ,,, )2 21 nyyy

indep. lin. ,,, )3 21 nyyy nyyy ,,, 21

Fundamental set of solutions

21, yy

xe(x)y 31 xe(x)y 3

2

These functions are solutions for the DE 0'9''' yy

Fund. Set of sol. ? 321 ,, yyy

13 (x)y

General Solution for Homog. DE

Theorem 4.5

THEN DE, theof solutions ofset fund. be Let 21 (x), y(x),(x),yy n

(x) yc(x)yc(x)ycy nn 2211

Is the general solution for the DE.

0)(')()()( 01)1(

1)(

yxayxayxayxa nn

nn

constants are where 21 n,c, ,cc

:Examplexe(x)y 3

1 xe(x)y 32

These functions are solutions for the DE 0'9''' yy

Find the general sol? 321 ,, yyy

13 (x)y

Given is a sol for 0'9''' yygeneral sol means what??

x(x)y 3sinh44

0)( yL

How to solve Homog. DE

Step 1

Step 2

Given a homg DE:

Find n-lin. Indep solutions

The general solution for the DE is

0)(')()()( 01)1(

1)(

yxayxayxayxa nn

nn

(x), y(x),(x),yy n21

(x) yc(x)yc(x)ycy nn 2211What is missing

0)( yL