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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

• 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

  