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

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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

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