# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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Find the general solution of

x

'0 1 1'

1 0 1| x.

.1 1 0>

(26)

n

Chapter 7. Systems of First Order Linear Equations

Observe that the coefficient matrix is real and symmetric. The eigenvalues and eigenvectors of this matrix were found in Example 5 of Section 7.3, namely,

A = -1,

r, = 2,

a = -1;

?(3) =

(27)

(28)

1

Hence a fundamental set of solutions of Eq. (26) is

x(1)(t) =

^2t

x(2)(t) =

1

x(3)(t) =

(29)

1

and the general solution is

1

x = C1 I 1 I e2t + c

0 I e + c3

1 I e-

(30)

1

1

This example illustrates the fact that even though an eigenvalue (r = —1) has multiplicity 2, it may still be possible to find two linearly independent eigenvectors ?(2) and ?(3) and, as a consequence, to construct the general solution (30).

The behavior of the solution (30) depends critically on the initial conditions. For large t the first term on the right side of Eq. (30) is the dominant one; therefore, if c1 = 0, all components of x become unbounded as t ^ro. On the other hand, for certain initial points c1 will be zero. In this case, the solution involves only the negative exponential terms and x ^ 0 as t ^ro. The initial points that cause c1 to be zero are precisely those that lie in the plane determined by the eigenvectors ?(2) and ?(3) corresponding to the two negative eigenvalues. Thus, solutions that start in this plane approach the origin as t ^ ro, while all other solutions become unbounded.

0

1

0

t

1

e

e

1

0

If some of the eigenvalues occur in complex conjugate pairs, then there are still n linearly independent solutions of the form (23), provided that all the eigenvalues are different. Of course, the solutions arising from complex eigenvalues are complexvalued. However, as in Section 3.4, it is possible to obtain a full set of real-valued solutions. This is discussed in Section 7.6.

More serious difficulties can occur if an eigenvalue is repeated. In this event the number of corresponding linearly independent eigenvectors may be smaller than the multiplicity of the eigenvalue. If so, the number of linearly independent solutions of the form ?ert will be smaller than n. To construct a fundamental set of solutions it is then necessary to seek additional solutions of another form. The situation is somewhat analogous to that for an nth order linear equation with constant coefficients; a repeated root of the characteristic equation gave rise to solutions of the form ert, tert, t2ert,.... The case of repeated eigenvalues is treated in Section 7.8.

Finally, if A is complex, then complex eigenvalues need not occur in conjugate pairs, and the eigenvectors are normally complex-valued even though the associated

7.5 Homogeneous Linear Systems with Constant Coefficients

381

PROBLEMS

eigenvalue may be real. The solutions of the differential equation (1) are still of the form (23), provided that the eigenvalues are distinct, but in general all the solutions are complex-valued.

In each of Problems 1 through 6 find the general solution of the given system of equations and describe the behavior of the solution as t ^ to. Also draw a direction field and plot a few trajectories of the system.

1. x' =

3. x' =

5. x' =

3 -2

2 -2

2

3 --12

2

12

> 2. x' =

> 4. x'

> 6. x' =

1 -2

3 -4

4 -2_

' 5 3

4 4

3 5

^ 4 4

In each of Problems 7 and 8 find the general solution of the given system of equations. Also draw a direction field and a few of the trajectories. In each of these problems the coefficient matrix has a zero eigenvalue. As a result, the pattern of trajectories is different from those in the examples in the text.

7. x'

4 -3

86

3 6

12

In each of Problems 9 through 14 find the general solution of the given system of equations.

9. x' =

11. x'

13. x'

1 1 2

1 2 1

2 1 1

1

2 1

8 5

10. x'

12. x'

1 -1 | x

2 2 + i\

-1 - - i)

3 2 4\

2 x

2

0

4 2 3/

1 -1 4'

3 2 -1

2 1 -1

In each of Problems 15 through 18 solve the given initial value problem. Describe the behavior of the solution as t .

15. x'

5 -1

,3 1

x(0) =

2

1

16 x'=L5 4)x x(0)=(3

17. x'

18. x' =

1 1

0

x(0) = ²0

x(0) = I 5

x

x

x

x

x

x

x

x

382

Chapter 7. Systems ofFirst Order Linear Equations

19. The system tx' = Ax is analogous to thesecond order Eulerequation(Section5.5). Assuming that x = ?f, where ? is a constant vector, show that ? and r must satisfy (A — rI) ? = 0 in order to obtain nontrivial solutions of the given differential equation.

Referring to Problem 19, solve the given system of equations in each of Problems 20 through

23. Assume that t > 0.

In each of Problems 24 through 27 the eigenvalues and eigenvectors of a matrix A are given. Consider the corresponding system x' = Ax.

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