ODE S C HAPTER 3. H IGHER O RDER L INEAR

C ^HAPTER 3. H ^IGHER O ^RDER L ^INEAR ODE ^S

2019.4

서울대학교

조선해양공학과

서유택

3.1 Homogeneous Linear ODEs





(Standard Form)





  ⁿ _n 1   ^{  n 1} 1   ' 0    

y  p _ x y ^   p x y  p x y  r x

 ^{, , ', ,}   ⁿ  ⁰

F x y y y 

 Homogeneous Linear ODE: Superposition Principle, General Solution

 Theorem 1 Fundamental Theorem for the Homogeneous Linear ODE

For a homogeneous linear ODE, sums and constant multiples of solutions on some open interval I are again solutions on I.

3.1 Homogeneous Linear ODEs

 Definition General Solution (일반해), Basis (기저), Particular Solution (특수해)

A general solution of an ODE on an open interval I is a solution on I of the form arbitrary

y = c ₁ y ₁

_n y _n

₁

_n

where y

₁

_n

that is, these solutions are linearly independent on I.

A particular solution of the equation on I is obtained if we assign specific values to the n constants y

₁

_n

₁

_n .

3.1 Homogeneous Linear ODEs

 Definition Linear Independence and Dependence (1차독립과 1차종속)

n functions y ₁

_n

k ₁ y ₁

_n y _n

₁

_n

₁

_n

These functions are called linearly dependent on I if this equation also holds on I

for some k ₁ , …, k _n not all zero.

 Ex. 1 Linear Dependence

Show y ₁ = x ³ , y ₂ = -2x, y ₃ = 3x ³ are linearly dependent on any interval.

3.1 Homogeneous Linear ODEs

 Theorem 2 Existence and Uniqueness Theorem for Initial Value Problems

If the coefficients p

₀

_n-1

₀

 Initial Value Problem (초기값문제). Existence and Uniqueness



  ⁿ _n 1   ^{  n 1} 1   ' 0   0

y  p _ x y ^   p x y  p x y 

y x  

 K y x

, '  

 K

, , y ^{ }

  x

 K

 Ex. 2 Solve the following initial value problem on any open interval I on

the positive x-axis containing x = 1.

Step 1 General solution of the homogeneous ODE

Corresponding general solution of the homogeneous ODE:

y _h = c ₁ x + c ₂ x ² + c ₃ x ³

Step 2 Particular Solution

Answer y _h

²

³

    

1 2 3 1 6 6 0 1, 2, 3 y  x

 m m  m   m m   m    m 

3.1 Homogeneous Linear ODEs

 Initial Value Problem for a Third-Order Euler-Cauchy Equation

4 )

1 ( ,

1 ) 1 ( , 2 ) 1 ( , 0 6

6 3

²

3 y

x y

x y

y

y

y

y

x

4 6

2 )

1 (

1 3

2 )

1 (

2 )

1 (

3 2

3 2

1

3 2

1







 







 









c c

y

c c

c y

c c

c y

1 2, 2 1, 3 1

c c c

    

3.1 Homogeneous Linear ODEs

 Linear Independence of Solutions. Wronskian

 

     

1 2

1 2

1

1 1 1

1 2

' ' '

, ,

n

n n

n n n

n

y y y

y y y

W y y

y ^ y ^ y ^



 Theorem 3 Linear Dependence and Independence of Solutions

Let the ODE have continuous coefficients p

₀

_n-1

Then n solutions y

₁

_n

if and only if their Wronskian is zero for some x = x ₀ in I.

Furthermore, if W is zero for x = x

₀

W is identically zero on I.

Hence if there is an x

₁

₁

_n

so that they form a basis of solutions of the equation on I.



1, x, x ² , x ³

3.1 Homogeneous Linear ODEs

 Initial Value Problem for a Third-Order Euler-Cauchy Equation

Q:?

3.1 Homogeneous Linear ODEs

 Theorem 5 General Solution Includes All Solutions

If the ODE has continuous coefficients p

₀

_n-1

where y

₁

_n

₁

_n

 

 

 

Y x  C y x   C y x

 A General Solution Includes All Solutions

 Theorem 4 Existence of a General Solution

If the coefficients p

₀

_n-1

 nth-order homogeneous linear ODEs with constant coefficients:

 We try .

 Characteristic equation (특성방정식):

 General Solutions

 Distinct Real Roots: all the n roots λ ₁

_n



constitute a basis.

 Simple Complex Roots: λ = γ±iω

 two corresponding linearly independent solutions are

1 2

y  e ^{ x} cos  x y ,  e ^{ x} sin  x

3.2 Homogeneous Linear ODEs with Constant Coefficients

    ¹

1 1 ' 0 0

n n

y  a n _ y ^   a y  a y 

1

1 1 0 0

n n

a n a a

  _  ^     

1

1 ^x , , _n ^{n x} y  e ^ y  e ^

y  e ^{ x}

 General Solutions

 Multiple Real Roots: λ is a real root of order m.

 m corresponding linearly independent solutions are

e ^λx , xe ^λx , x ² e ^λx , ···, x ^m-1 e ^λx .

 Multiple Complex Roots: λ = γ±iω are double roots.

 corresponding linearly independent solutions are

e ^γx cos ωx, e ^γx sin ωx, xe ^γx cos ωx, xe ^γx sin ωx.

3.2 Homogeneous Linear ODEs with Constant Coefficients

 Ex 2. Solve the initial value problem.

3.2 Homogeneous Linear ODEs with Constant Coefficients

 Simple Complex Roots. Initial Value Problem

299 0

11 0

4 0

0 100

100         

 

  y y y y ( ) y ( ) y ( )

y

3 2

100 100 0

          1, 10i

1 1

1

cos10 sin10

10 sin10 10 cos10 100 cos10 100 sin10

x x

x

y c e A x B x

y c e A x B x

y c e A x B x

  

   

   

1 1 1

4 10 11

100 299

c A

c B

c A

 

 

  

101 A  303,   A 3

1

B 

1

c 

3cos10 sin10

y e x x x

   

 Ex. Solve the ODE.

3.2 Homogeneous Linear ODEs with Constant Coefficients

 Real Double and Triple Roots

Q:?

2 0

y iv  y    y



n 계 비제차 선형 상미분방정식)



_h

_p

 y _h

₁ y ₁

_n y _n

 y _p



 

  ^{ }

  '

    ,   0 y  p

x y

  p x y  p x y  r x r x 

 

  ^{ }

  '

  0 y  p

x y

  p x y  p x y 

 

  ^{ }

  '

   

y  p

x y

  p x y  p x y  r x

3.3 Nonhomogeneous Linear ODEs

 

  ^{ }

  '

   

y  p

x y

  p x y  p x y  r x

y x  

 K

, ' y x  

 K

, , y ^{ }

  x

 K

3.3 Nonhomogeneous Linear ODEs

 Method of Undetermined Coefficients (미정계수법)

 

  ^{ }

  '

    ,   0 y  p

x y

  p x y  p x y  r x r x 

 Choice Rules for the Method of Undetermined Coefficients

a. Basic Rule. If r(x) is one of the functions in the first column in Table 2.1, choose y _p in

the same line and determine its undetermined coefficients by substituting y _p and its

derivatives into y ⁽ⁿ⁾ +a _n-1 y ^(n-1) + ··· + a ₁ y ʹ + a ₀ y = r(x).

3.3 Nonhomogeneous Linear ODEs

 Choice Rules for the Method of Undetermined Coefficients

b. Modification Rule. If a term in your choice for y _p is a solution of the

homogeneous ODE corresponding to y ⁽ⁿ⁾ +a _n-1 y ^(n-1) + ··· + a ₁ y ʹ + a ₀ y = r(x) then multiply this term by x ^k , where k is the smallest positive integer such that this term times x ^k is not a solution of the homogeneous ODE.

 Method of Undetermined Coefficients (미정계수법)

 

  ^{ }

  '

    ,   0

y  p

x y

  p x y  p x y  r x r x 

3.3 Nonhomogeneous Linear ODEs

 Choice Rules for the Method of Undetermined Coefficients

c. Sum Rule. If r(x) is a sum of functions in the first column of Table 2.1, choose for y _p the sum of the functions in the corresponding lines of the second column.

 Method of Undetermined Coefficients (미정계수법)

 

  ^{ }

  '

    ,   0

y  p

x y

  p x y  p x y  r x r x 



3.3 Nonhomogeneous Linear ODEs

 Simple Complex Roots. Initial Value Problem

47 0

3 0

3 0

30 3

3           

  y y y e ^ y ( ) y ( ) y ( )

y ^x

(not satisfied)

(from modification rule)



3.3 Nonhomogeneous Linear ODEs

 Simple Complex Roots. Initial Value Problem

47 0

3 0

3 0

30 3

3           

  y y y e ^ y ( ) y ( ) y ( )

y ^x

3.3 Nonhomogeneous Linear ODEs





W _j

^T

When n = 2, this becomes identical with

If it starts with f (x)y, divide first by f (x).

     

       

   

1 1

n

p n

W x W x

y x y x r x dx y x r x dx

W x W x

    

1 1

1 2

2 2

2 1

2 1

2 1

1 0 1

0 y

y W y

y y W y

y y

y

W y 

 



 

 

  , ,

 

p

y r y r

y x y dx y dx

W W

    

 Ex. 2 Solve the nonhomogeneous Euler-Cauchy equation.

Step 1 General solution of the homogeneous ODE

Corresponding general solution of the homogeneous ODE:

y _h

₁ x + c ₂ x ²

₃ x ³

 y

₁

₂

²

₃

³ Step 2 Determinants

3 2 4

''' 3 '' 6 ' 6 ln

x y

x y

xy

y

x x

    

1 2 3 1 6 6 0 1, 2, 3 y  x

 m m  m   m m   m    m 

2 3 2 3 3 2

2 3 2 4 2 3 2

1 2 3

0 0 0

1 2 3 2 , 0 2 3 , 1 0 3 2 , 1 2 0

0 2 6 1 2 6 0 1 6 0 2 1

x x x x x x x x x

W x x x W x x x W x x W x x

x x x

        

3.3 Nonhomogeneous Linear ODEs

 Ex. 2 Solve the nonhomogeneous Euler-Cauchy equation

Step 3 Integration

Here, since it starts with x ³ y, divide first by x ³ . Then, r(x) = x ³ lnx/x ⁴ = xlnx.

3 2 4

''' 3 '' 6 ' 6 ln

x y

x y

xy

y

x x

3 4 3 2

1 2 3

2 , , 2 ,

W

x W

x W

x W

x

2 3 4

2 3 4

1 2 3

1 1 11

ln ln ln ln

2 2 6 6

1 11

ln

6 6

p

y x x x xdx x x xdx x x xdx x x x

y c x c x c x x x

 

      

 

 

       

 

  

3.3 Nonhomogeneous Linear ODEs

     

       

   

1 1

n

p n

W x W x

y x y x r x dx y x r x dx

W x W x

    

y ₁

₂

²

₃

³