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Chapter 2 Second Order Linear ODEs 2.1 Homogeneous Linear Equations of 2nd Order A 2nd order linear equation: ) ( ) ( ' ) ( " x r y x q y x p y = + + (1) is called nonhomogeneous. ) 0 ) ( ( x r (1) 0 ) ( ' ) ( " = + + y x q y x p y (2) is called homogeneous.

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Page 1: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Chapter 2 Second Order Linear ODEs 2.1 Homogeneous Linear Equations of 2nd Order

A 2nd order linear equation:

)()(')(" xryxqyxpy =++(1)

is called nonhomogeneous. )0)(( ≠xr(1)

0)(')(" =++ yxqyxpy(2)

is called homogeneous.

Page 2: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

For instance a nonhomogeneous lin. ODE

xeyy x cos25 −=+′′

A homogeneous lin. ODE

0' =++′′ xyyyxA nonlinear ODE

02 =′+′′ yyy

Page 3: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Superposition Principle for the Homogeneous Linear ODEs:

Ex. 1 xyxy sin and cos ==are solutions of 0" =+ yy

Then xcxcy sincos 21 +=

is also a solution of the above Eq.

---superposition or linearity principle

Page 4: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Theorem 1. If 1y and 2y are solutions of (2), then 2211 ycycy += is a solution of (2).

Proof: 0)()( 111 =+′+′′ yxqyxpy0)()( 222 =+′+′′ yxqyxpy

We have

0)()()()'()"(

)()(

22221111

221122112211

=+′+′′++′+′′=+++++=

+′+′′

qyypycqyypycycycqycycpycyc

yxqyxpy

Page 5: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Ex. 2 Verify by substitution that xyxy sin1 and cos1 +=+=

are solutions of .1" =+ yy

xxyxy sincos2or )cos1(5 ++=+=

are not solutions of !1" =+ yy

Page 6: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Ex. 3 Verify that 1 and 2 == yxy

are solutions of the nonlinear ODE 0=′−′′ yxyy

22 1or xyxy +=−=

are not solutions of 0=′−′′ yxyy

Page 7: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

IVP. Basis. General Solution Initial conditions: (4) 1000 )( ,)( KxyKxy =′=

(4) with (2) is called IVP for the second order homogenous linear ODE.

Page 8: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Ex. 4 Solve the initial value problem .5.0)0(' ,0.3)0( ,0" −===+ yyyy

Soluton. Step 1 xcxcy sincos 21 +=

is the general solution Step 2 particular solution

5.0)0(' and 0.3)0( 21 −==== cycy

xxy sin5.0cos0.3 −=

Page 9: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1
Page 10: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Definition 1. Two solutions 1y and 2y

of (2) are called a basis of (2) iff constant/ 21 ≠yy

Definition 2. Two solutions 1y and 2y

of (2) are a basis of (2), then

2211 ycycy +=

is called a general solution of (2).

Page 11: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

A particular solution of (2) is obtained if we assign specific values to 1c and 2cin the general solution.

We call linearly independent if 21 and yy0 ,0 0 212211 ==⇒=+ ccycyc(7)

Definition 1’. A pair solutions 21 and yyof (2) are called a basis of (2) iff 21 and yy

are linear independent.

Page 12: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Ex. 5 cos x and sin x in Ex. 4 form a basis of solutions of y”+y=0 for all x. y=3.0cos x-0.5sin x is a particular solution.

Page 13: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Ex.6 Verify xx ee − and are solutions of .0" =−yy Then solve the IVP

2)0(' ,6)0( −== yy

Solution. ( ) ( ) 0 and 0 =−″

=−″ −− xxxx eeee

xx ececy −+= 21 is general sol.

4 ,22)0(' and 6)0(

21

2121

==⇒−=−==+=

ccccyccy

xx eey −+= 42 is particular sol.

Page 14: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

A nontrivial solution of (2) 1y is obtained by some method, how to get the other 2y

to form a basis of (2)?

Set 12 uyy = then we have

112 '' uyyuy +=′ and

1112 "''2"" uyyuyuy ++= This gives

0)''("''2"

111

111

=+++++

quyuyyupuyyuyu

hence

Page 15: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

.0')'2(",0)'"(')'2("

111

111

111

=++=+++

++

upyyyuqypyyu

upyyyu

Let "' ,' uUuU == then

.0)/'2(U' 11 =++ Upyy

Page 16: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

dxpey

U

dxpyU

dxpyy

∫−=

−−=

+−=

21

1

11

1

,||ln2||ln

,)/'2(U

dU

dxuuU ∫== U ,' dxUyuyy ∫== 112

(?)/ 12 constdxUuyy ≠== ∫form a basis of (2). 21 and yy

Page 17: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Ex. 7 Find a basis of solutions of

.0'")( 2 =+−− yxyyxxSolution. xy =1 is one of the solution

Set uxuyy == 12 then we have uxuy +=′ '2 and

'2""2 uxuy += This gives

0)'()'2")(( 2 =++−+− uxuxuxuxuxx

hence

Page 18: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

0')2(")( 2 =−+− uxuxx Let 'uv =

0)2(')( 2 =−+− vxvxx

dxxx

dxxx

xvdv

−=

−−

−=2

112

2

|||1|ln||ln2|1|ln||ln

xxxxv −

=−−=

∫ +==−=−

=x

xvdxuxxx

xv 1||ln ,11122

Page 19: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Finally, we have

.1||ln2 +== xxuxy

,1 xy =with

They form a basis of the solutions.

Page 20: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

2.2 2nd-Order Homogeneous Equations with Constant Coefficients

0'" =++ byayy(1) kxey −= is a solution of .0' =+kyy

To try as a solution of (1) the function xey λ=(2)

xey λλ=' and xey λλ2"= (1) becomes

0)( 2 =++ xeba λλλ

Page 21: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

We got a characteristic equation of (1)

02 =++ baλλ(3)

2/)4( 22,1 baa −±−=λ(4)

xey 11

λ= and xey 2

2λ=

are solutions of (1).

Page 22: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

042 >− ba042 =− ba

042 <− ba

Case I two real roots if

Case II a real double root if

Case III complex roots if

There are 3 cases for the roots.

Page 23: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Case I. Two Distinct Real Roots xey 1

1λ= and

xey 22

λ=

constant/ )(21

21 ≠= − xeyy λλ

The general solution is

xx ececy 2121

λλ +=

Page 24: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Ex. 1 in Ex. 6 of Section 2.1 .0" =−yySolution. The characteristic equation is

012 =−λIt has the roots 1±=λ

Hence the general solution xx ececy −+= 21

Page 25: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Ex. 2 Solve the IVP .5)0(' ,4)0( ,02'" −===−+ yyyyy

Solution. The characteristic equation is 022 =−+ λλ

It has the roots 2 ,1 21 −== λλ

Hence the general solution xx ececy 2

21−+=

Page 26: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

xx ececy 221 2' −−=

According to the initial condition:

3 ,152)0(' and 4)0(

21

2121

==⇒−=−==+=

ccccyccy

Page 27: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Case II. Real Double Root

042 =− ba 2/21 a−=== λλλhence only one solution xaey )2/(

1−=

12 uyy =Set 112 '' uyyuy +=′

1112 "''2"" uyyuyuy ++=

0)''("''2"

111

111

=+++++

buyuyyuauyyuyu

Page 28: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

.,0"

)2/' (since 0",0')'2("

,0)'"(')'2("

21

111

111

111

111

cxcuu

yayyuuayyyu

byayyuuayyyu

+==

−===++=+++

++

xu =

xaxexyy )2/(12

−==

Page 29: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

0/ 12 ≠= xyy

The general solution is xaecxcy )2/(

21 )( −+=

Warning. (7) will not be a solution of (1) for the simple root of (4).

(7)

Page 30: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Ex. 3 Solve 016'8" =++ yyy

Solution. The characteristic equation is 01682 =++ λλ

It has the double root 4−=λand hence the general solution is

xecxcy 421 )( −+=

Page 31: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Ex. 4 Solve the initial value problem 1)0(' ,3)0( ,04'4" ===+− yyyyy

Solution. The characteristic equation is

0442 =+− λλIt has the double root 2=λand hence the general solution is

xexccy 221 )( +=

Page 32: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

xx ecexccy 22

221 )(2' ++=

5 ,3,12)0('

,3)0(

21

21

1

−===+=

==

ccccy

cy

and the answer is

xexy 2)53( −=

Page 33: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1
Page 34: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Case III. Complex Roots

If (2) has a pair of complex roots 4/ ,2/ , 2

2,1 abai −=−=±= ωαωαλ

xiaey )2/(1

ω+−= and xiaey )2/(2

ω−−=

are solutions of (1).

To find the real solutions, we can obtain from (5) that

Page 35: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

)sin(cos

),sin(cos2/)2/(

2

2/)2/(1

xixeeyxixeey

axxia

axxia

ωω

ωωω

ω

−==

+==−−−

−+−

and hence that xe ax ωcos2/− and

xe ax ωsin2/− are solutions of (1).

The corresponding general solution is

)sincos(2/ xBxAey ax ωω += −(9)

Page 36: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Ex. 5. Solve the initial value problem 3)0(' ,0)0( ,004.9'4.0" ===++ yyyyy

Solution. The characteristic equation is

004.94.02 =++ λλIt has the roots ,32.02,1 i±−=λand a general solution is

)3sin3cos(2.0 xBxAey x += −

Page 37: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

The 1st initial condition gives

0)0( == Ay

It concludes that .3sin2.0 xBey x−=

)3cos33sin2.0(' 2.0 xxBey x +−= −

From this and the 2nd initial condition we get .133)0(' =⇒== BBy The answer is

xey x 3sin2.0−=

Page 38: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1
Page 39: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Ex. 6 A general solution of 0 where,0" 2 ≠=+ ωω yy

is

xBxAy ωω sincos +=

Page 40: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Ex. 7 Solve the boundary value problem

3)2/( ,3)0( ,0" −===+ πyyyy

Solution. The general solution is known as xBxAy sincos +=

So we have 3)2/( ,3)0( −==== ByAy π

The answer is xxy sin3cos3 −=

Page 41: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1
Page 42: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

2.5 Euler-Cauchy Equation

0'"2 =++ byaxyyx(1)

is called the Euler-Cauchy equation. Let mxy = is a solution of (1), we find that

0)1( 122 =++− −− mmm bxaxmxxmmx or

0)1(2 =+−+ bmam(3)

(3) is called the characteristic equation of (1).

Page 43: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Case I. Two Distinct Real Roots If (3) has two distinct real roots 2,1mwe get a basis of solutions 21

21 and mm xyxy ==

constant/ 2121 ≠= −mmxyy

and a corresponding general solution of (1) is as follows

2121

mm xcxcy +=(4)

Page 44: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Ex. 1 Solve the Euler-Cauchy equation 05.0'5.1"2 =−+ yxyyx

Solution. The characteristic equation is 05.05.02 =−+ mm

?) 05.05.1(not 2 =−+ mmThe roots are m=0.5 and -1, this gives

xy =1 and xy /12 =and the general solution

0 ,/21 >∀+= xxcxcy

Page 45: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Case II. Real Double Root

If (3) has a double root, then 2/)1(2,1 am −=

hence only one solution .2/)1(1

axy −=

Set ,12 uyy = then we have

112 '' uyyuy +=′

1112 "''2"" uyyuyuy ++= This gives

0)''()"''2"(

111

1112

=+++++buyuyyuax

uyyuyux

Page 46: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

hence

.ln||ln|'|ln ,'"

)2/)1(' (since "

,0')'2("

,0)'"(

')'2("

11

112

1112

1112

1112

xuxuuxu

yaxyxu'yyxu

uayxyxyxubyaxyyxu

xuayxyyxu

=−=−=

−=−=

=++

=+++

++

Page 47: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Thus ,ln12 xyy = where constxyy ≠= ln/ 12

The general solution is

2/)1(21 )ln( axxccy −+=

Page 48: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Ex. 2 Solve 09'5"2 =+− yxyyx

Solution. The characteristic equation 0962 =+− mm

has the double root m=3

Hence the general solution is

321 )ln( xxccy +=

Page 49: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Case III. Complex Conjugate Roots

If (3) has a pair of complex roots im νµ ±=2,1

then xii exxxy ln2,1

νµνµ ±± == are solutions of (1). To find the real solutions, we have

)]lnsin()ln[cos(

)],lnsin()ln[cos(

2

1

xixxyxixxy

νν

ννµ

µ

−=

+=

and hence that )lncos( xx νµ and

)lnsin( xx νµ are solutions of (1).

Page 50: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

The corresponding general solution is

)]lnsin()lncos([ xBxAxy ννµ +=

Ex. 3 Solve 004.16'6.0"2 =++ yxyyx

Solution. The characteristic equation 004.164.02 =+− mm

has two complex conjugate roots

im 42.02,1 ±=

Page 51: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Hence the general solution is

)]ln4sin()ln4cos([2.0 xBxAxy +=

Page 52: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

2.6 Existence and Uniqueness Theory Wronskian A general homogeneous linear equation

0)(')(" =++ yxqyxpy(1)

with continuous coefficients. The existence of a general solution

2211 ycycy +=and two initial conditions

1000 )(' ,)( KxyKxy ==

(2)

(3)

Page 53: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

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

Ix ∈0

Wronski determinant (or Wronskian) of two solutions of (1) as and 21 yy

122121

2121 ''

''),( yyyy

yyyy

yyW −==

Page 54: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

We call linearly independent on I if 21 and yy0 ,0 on 0 212211 ==⇒=+ ccIycyc(4)

We call linearly dependent if 21 and yy

(4) holds for constants c1 and c2 not both 0.

Page 55: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Theorem 2. If p(x) and q(x) are continuous on some open interval I. Then two solutions of (1) on I are linearly dependent iff and 21 yy

. ,0))(),(( 00201 IxxyxyW ∈=

Furthermore, if . ,0))(),(( 00201 IxxyxyW ∈=

; ,0))(),((then 21 IxxyxyW ∈∀≡ hence if

, ,0))(),(( 11211 IxxyxyW ∈≠ then and 21 yy

are linearly independent on I.

Page 56: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Proof: (a) If and 21 yy are linearly dependent

on I, then there exists k, s.t. 21 kyy = and

0''''''

),( 222222

22

21

2121 ≡−=== kyyyky

ykyyky

yyyy

yyW

(b) Conversely, we assume that

IxxyxyW ∈= 00201 ,0))(),(( and show that

and 21 yy are linearly dependent on I.

Page 57: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

We consider the linear equations

0)(')(',0)()(

022011

022011

=+=+

xykxykxykxyk

in the unknowns and 21 kk

(7)

(7) is homogeneous.

Its determinant is exactly 0))(),(( 0201 =xyxyW

hence the system has a nontrivial solution and 21 kk

Page 58: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Then we have a function )()()( 2211 xykxykxy +=

By Theorem 1 in Sec. 2.1, y(x) is a solution of (1) on I. We know that 0)(' ,0)( 00 == xyxy

It concludes by Theorem 1 that . ,0 Ixy ∈∀≡

Ixxykxyk ∈∀=+ ,0)()( 2211

this means linear dependence of and 21 yyon I.

Page 59: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

(c) If 0))(),(( 0201 =xyxyW then and 21 yyare linear dependent, and then by (a),

.0),( 21 ≡yyW Hence 0))(),(( 1211 ≠xyxyW

implies linear independence of and 21 yy

Page 60: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Ex. 1 xy ωcos1 = and xy ωsin2 =

are solutions of .0" 2 =+ yy ω The Wronskian is

ωωωωωωω

=−

=xx

xxyyW

cossinsincos

),( 21

They are linearly independent iff .0≠ω0=ωIf then ,02 =y

hence they are linear dependence.

Page 61: Chapter 2 Second Order Linear ODEs - University of Macaufstitl/Calculus2014/Chapter2_PPT.pdf · A particular solution of (2) is obtained if we . assign specific values to . c. 1

Ex. 2 A general solution of 0'2" =+− yyy

on any interval is .)( 21xexccy +=

The Wronskian is

0)1(

),( 2 ≠=+

= xxx

xxxx e

exexee

xeeW

Thus xe and xxe are linearly independent.

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

Proof. By Theorem 1, (1) has solutions and 21 yy on I satisfying the initial conditions:

1)(' ,0)( 0101 == xyxy and 0)(' ,1)( 0202 == xyxy

Then and 21 yy are linearly independent on I. Thus they form a basis of (1) and then

2211 ycycy += is a general solution of (1).

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Theorem 4 If p(x) and q(x) of (1) are continuous on an open interval I, then for every solution y=Y(x) of (1) is of the form

)()()( 2211 xyCxyCxY +=

where and 21 yy form a basis of (1) on I

1C 2Cand , are suitable constants.

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Proof. A general solution of (1) is given as .2211 ycycy += We choose the suitable

and 21 cc such that )(')(' ),()( 0000 xYxyxYxy ==

),()()( 0022011 xYxycxyc =+ )(')(')(' 0022011 xYxycxyc =+

The Wronskian of and 21 yy is nonzero,

(10)

such that (10) has a unique solution 11 Cc =

and .22 Cc =

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Thus a particular solution

)()()(* 2211 xyCxyCxy += of (1) satisfies (10). From this and the uniqueness theorem 1 we conclude that everywhere on I. )()(* xYxy =

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2.7 Nonhomogeneous Equations

The nonhomogeneous linear equation

)()(')(" xryxqyxpy =++(1)

Its corresponding homogeneous equation is

0)(')(" =++ yxqyxpy(2)

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Theorem 1. (a) If are solutions of (1) on I, then is a solution of (2) on I. (b) If y is a solution of (1) on I, and Y is a

solution of (2) on I, then is a solution of (1) on I.

and 21 yy21 yy −

Yy +

Proof. (a) Denote that then ,21 yyY −=

0)()()'"()'"()()'()"(

'"

222111

212121

≡−=++−++=−+−+−=

++

xrxrqypyyqypyy

yyqyypyyqYpYY

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(b) Denote that then ,* yYy +=

)()(0)'"()'"()()'()"(

**)'("*)(

xrxrqypyyqYpYYyYqyYpyY

qyypy

=+=+++++=+++++=

++

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Definition. A general solution of (1) on I is )()()( xyxyxy ph +=(3)

where 2211 ycycyh += is a general solution of (2), and )(xy p is any solution of (1).

A particular solution of (1) on I is a solution obtained from (3) by assigning specific values to the arbitrary constants in and 21 cc

.2211 ycycyh +=

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Theorem 2. Suppose that the coefficients and r(x) in (1) are continuous on I. Then every solution of (1) on I is obtained from (3) by assigning suitable values to . and 21 cc

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Proof. Let y*(x) be any solution of (1) on I. Let (3) be any general solution of (1) on I. Then is a solution of (2) by Theorem 1(a). By theorem 4 in Sec. 2.6, Y is obtained by assigning suitable values to the arbitrary constants From this and the statement follows.

pyyY −= *

. and 21 cc,* pyYy +=

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Rules for the method of undetermined coefficients A.Basic Rule. Choosing in the Table. B. Modification Rule. Multiplying by

where j=1 or 2, if or is a solution of the corresponding homogeneous equation (or double roots).

C.Sum Rule. If r(x) is a sum of functions in the table, then is chosen as the sum of corresponding functions.

pyjx

xeγ xe x ωα cos

py

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Ex. 1. Solve the equation

Solution. Solve the corresponding homogenous equation. We have

5.1)0(',0)0(,001.0" 2 ===+ yyxyy

,012 =+λ i±=2,1λ and then .sincos xBxAyh +=

We find that 0 is not a root of the characteristic equation. So

012

2 kxkxkyp ++= is chosen. Then

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212 2",2 kykxky pp =+=′

.001.0)(2 201

222 xkxkxkk =+++

Compare with the coefficients above, we have

.02 ,0 ,001.0 2012 =+== kkkk Thus

.002.02 ,0 ,001.0 0012 −=−=== kkkk

Hence .002.0001.0 2 −= xyp And

002.0001.0sincos 2 −++= xxBxAy

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Setting x=0, y(0)=A-0.002=0, hence A=0.002. y’(0)=B=1.5. Thus gives the answer

002.0001.0sin5.1cos002.0 2 −++= xxxy

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Ex. 2. Solve the initial value problem 0)0(' ,1)0( ,1025.2'3" 5.1 ==−=++ − yyeyyy x

Solution. The characteristic equation is .025.232 =++ λλ It has the double root

.5.1−=λ and a general solution is

.)( 5.121

xexccy −+= At first, xp Cey 5.1−=

by the table. According to the rule, we choose

.5.12 xp eCxy −= Hence x

p exxCy 5.12 )25.1(' −+−=

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.)25.262(" 5.12 xp exxCy −+−= Substitution gives

xxxx eeCxexxCexxC 5.15.125.125.12 1025.2)5.12(3)25.262( −−−− −=+−++−

,102 5.15.1 xx eCe −− −= 5−=CThus gives the general solution is

xexxccy 5.1221 )5( −−+=

,1)0( 1 −== cy)5.7105.15.1(' 2

225.1 xxxccey x +−−−= −

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5.105.1)0(' 22 =⇒=−= ccyThis gives the answer xexxy 5.12 )55.11( −−+=

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Ex. 3 Solve the initial value problem

08.40)0(' ,16.0)0( ,10sin19010cos405'2" 2/

==−+=++

yyxxeyyy x

Solution. The characteristic equation is .0522 =++ λλ It has the roots

,212,1 i±−=λ and a general solution is

).2sin2cos( xBxAey x += −

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We choose .10sin10cos2/ xMxKCey x

p ++=

.10cos1010sin102/' 2/ xMxKCey xp +−=

.10sin10010cos1004/" 2/ xMxKCey xp −−=

Substitute this into the given equation.

16.0)514/1( 2/2/ =⇒=++ CeCe xx

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.2,01909520,409520

.10sin19010cos4010cos)2095(10sin)2095(

10cos)520100(10sin)520100(

==⇒−=−−=−⇒

−=+−+−−=

+−−+++−

MKMKKM

xxxMKxKM

xKMKxMKM

This gives the general solution

10sin216.0)2sin2cos( 2/ xexBxAey xx +++= −

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From the initial conditions, we have

.1008.402008.02)0('10cos2008.0)2cos22sin('

016.016.0)0(2/

=⇒=++=+++−=

=⇒=+=−

BByxexBxBey

AAyxx

We thus obtain the answer

10sin216.02sin10 2/ xexey xx ++= −

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2.10 Solution by Variation of Parameters A general nonhomogeneous linear equation

)()(')(" xryxqyxpy =++(1) with continuous ).(),(),( xrxqxpMethod of variation of parameters,

dxrydxryy p ∫∫ +−=Wy

Wy 1

22

1(2) where 21, yy form a basis of the corresponding homogeneous equation of (1)

0)(')(" =++ yxqyxpy(3)

1221 '' yyyyW −= is the Wronskian of 21, yy

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Ex. 1. Solve xyy sec" =+Solution. A basis of the homogeneous equation is Thus .sin and cos 21 xyxy ==

.1)sin(sincoscos'' 1221 =−−=−= xxxxyyyyW

xxxxxdxxxxdxxx

dxrydxryyp

sin|cos|lncosseccossinsecsincos

Wy

Wy 1

22

1

+=+−=

+−=

∫∫∫∫

xxcxxcxxxxxcxcy

sin)(cos|)cos|ln( sin|cos|lncossincos

21

21+++=

+++=We then obtain the answer

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How to get the method? )()()( xyxyxy ph += with 2211 ycycyh +=

Let be a solution of (1). )()()()( 21 xyxvxyxuy p +=

2121 ''''' vyuyyvyuy p +++=0''Set 21 =+ yvyu then 21 ''' vyuyy p +=

2121 ""''''" vyuyyvyuy p +++=

Substituting to (1), and collecting terms

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⇒=+++++++

=+++++++

ryvyuqypyyvqypyyu

rvyuyqvyuypvyuyyvyu

21

222111

21

'212121

'''')'"()'"(

)()'()""''''(

ryvyu =+ 21 '''' with 0'' 21 =+ yvyu

So we have W

ryvW

ryu 12 ' ,' =−=

dxW

ryvdxW

ryu ∫∫ =−= 12 ' ,

dxrydxryy p ∫∫ +−=Wy

Wy 1

22

1

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Pages 103-104c

Continued

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Pages 103-104d