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Transcript of Boyce/DiPrima/Meade 11 ed, Ch 7.1: Introduction to Systems ... · PDF file Boyce/DiPrima/Meade...

  • Boyce/DiPrima/Meade 11th ed, Ch 7.1: Introduction to Systems

    of First Order Linear Equations

    Elementary Differential Equations and Boundary Value Problems, 11th edition, by William E. Boyce, Richard C. DiPrima, and Doug Meade ©2017 by John Wiley & Sons, Inc.

    • A system of simultaneous first order ordinary differential

    equations has the general form

    where each xk is a function of t. If each Fk is a linear

    function of x1, x2, …, xn, then the system of equations is said

    to be linear, otherwise it is nonlinear.

    • Systems of higher order differential equations can similarly

    be defined.

    ),,,(

    ),,,(

    ),,,(

    21

    2122

    2111

    nnn

    n

    n

    xxxtFx

    xxxtFx

    xxxtFx

    

    

    

  • Example 1

    • The motion of a certain spring-mass system from Section 3.7

    was described by the differential equation

    • This second order equation can be converted into a system of

    first order equations by letting x1 = u and x2 = u'. Thus

    or

    ¢¢u (t)+ 1

    8 ¢u (t)+ u(t) = 0

    ¢x1 = x2

    ¢x2 + 1

    8 x2 + x1 = 0

    ¢x1 = x2

    ¢x2 = -x1 - 1

    8 x2

  • Nth Order ODEs and Linear 1st Order

    Systems

    • The method illustrated in the previous example can be used

    to transform an arbitrary nth order equation

    into a system of n first order equations, first by defining

    Then

     )1()( ,,,,,  nn yyyytFy 

    )1(

    321 ,,,,  nn yxyxyxyx 

    ),,,( 21

    1

    32

    21

    nn

    nn

    xxxtFx

    xx

    xx

    xx

    

    

    

    

  • Solutions of First Order Systems

    • A system of simultaneous first order ordinary differential

    equations has the general form

    It has a solution on if there exists n functions

    that are differentiable on I and satisfy the system of

    equations at all points t in I.

    • Initial conditions may also be prescribed to give an IVP:

    ).,,,(

    ),,,(

    21

    2111

    nnn

    n

    xxxtFx

    xxxtFx

    

    

    )(,),(),( 2211 txtxtx nn   

    0

    0

    0

    202

    0

    101 )(,,)(,)( nn xtxxtxxtx  

    I :a < t < b

  • Theorem 7.1.1

    • Suppose F1,…, Fn and

    are continuous in the region R of t x1 x2…xn-space defined by

    and let the point

    be contained in R. Then in some interval

    (t0 – h, t0 + h) there exists a unique solution

    that satisfies the IVP.

     002010 ,,,, nxxxt 

    )(,),(),( 2211 txtxtx nn   

    ),,,(

    ),,,(

    ),,,(

    21

    2122

    2111

    nnn

    n

    n

    xxxtFx

    xxxtFx

    xxxtFx

    

    

    

    ¶F1 / ¶x1,...,¶F1 / ¶xn,...,¶Fn / ¶x1,...,¶Fn / ¶xn

    a < t < b,a1 < x1 < b1,...,an < xn < bn

  • Linear Systems

    • If each Fk is a linear function of x1, x2, …, xn, then the

    system of equations has the general form

    • If each of the gk(t) is zero on I, then the system is homogeneous, otherwise it is nonhomogeneous.

    )()()()(

    )()()()(

    )()()()(

    2211

    222221212

    112121111

    tgxtpxtpxtpx

    tgxtpxtpxtpx

    tgxtpxtpxtpx

    nnnnnnn

    nn

    nn

    

    

    

  • Theorem 7.1.2

    • Suppose p11, p12,…, pnn, g1,…, gn are continuous on an

    interval with t0 in I, and let

    prescribe the initial conditions. Then there exists a unique

    solution

    that satisfies the IVP, and exists throughout I.

    00

    2

    0

    1 ,,, nxxx 

    )(,),(),( 2211 txtxtx nn   

    )()()()(

    )()()()(

    )()()()(

    2211

    222221212

    112121111

    tgxtpxtpxtpx

    tgxtpxtpxtpx

    tgxtpxtpxtpx

    nnnnnnn

    nn

    nn

    

    

    

    I :a < t < b

  • Boyce/DiPrima/Meade 11th ed, Ch 7.2:

    Review of Matrices

    Elementary Differential Equations and Boundary Value Problems, 11th edition, by William E. Boyce, Richard C. DiPrima, and Doug Meade ©2017 by John Wiley & Sons, Inc.

    • For theoretical and computational reasons, we review results

    of matrix theory in this section and the next.

    • A matrix A is an m x n rectangular array of elements,

    arranged in m rows and n columns, denoted

    • Some examples of 2 x 2 matrices are given below:

     

        

        

    

    mnmm

    n

    n

    ji

    aaa

    aaa

    aaa

    a

    

    21

    22221

    11211

    A

     

      

    

     

      

     

      

     

    ii

    i CB

    7654

    231 ,

    42

    31 ,

    43

    21 A

  • Transpose

    • The transpose of A = (aij) is A T = (aji).

    • For example,

        

        

    

        

        

    mnnn

    m

    m

    T

    mnmm

    n

    n

    aaa

    aaa

    aaa

    aaa

    aaa

    aaa

    

    

    21

    22212

    12111

    21

    22221

    11211

    AA

      

      

     

      

     

      

     

      

     

    63

    52

    41

    654

    321 ,

    42

    31

    43

    21 TT

    BBAA

  • Conjugate

    • The conjugate of A = (aij) is A = (aij).

    • For example,

        

        

    

        

        

    mnmm

    n

    n

    mnmm

    n

    n

    aaa

    aaa

    aaa

    aaa

    aaa

    aaa

    

    

    21

    22221

    11211

    21

    22221

    11211

    AA

     

      

     

      

     

    443

    321

    443

    321

    i

    i

    i

    i AA

  • Adjoint

    • The adjoint of A is AT , and is denoted by A*

    • For example,

        

        

    

        

        

    mnnn

    m

    m

    mnmm

    n

    n

    aaa

    aaa

    aaa

    aaa

    aaa

    aaa

    

    

    21

    22212

    12111

    *

    21

    22221

    11211

    AA

     

      

     

      

     

    432

    431

    443

    321 *

    i

    i

    i

    i AA

  • Square Matrices

    • A square matrix A has the same number of rows and

    columns. That is, A is n x n. In this case, A is said to have

    order n.

    • For example,

        

        

    nnnn

    n

    n

    aaa

    aaa

    aaa

    

    21

    22221

    11211

    A

      