# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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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 —2

2

1 -2

? 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

x

3

V-1 —2 j

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

9. x' =

11. x'=

13. x' =

1i

i1

1 1 2

1 2 1

2 1 1

1

2 1

8 5

10. x' =

12. x' =

1 -11 x

2 2 + i\

—1 —1 — i)

3 2 4\

2 0 2 I x

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 -> oo.

15. x' =

16. x' =

17. x' =

18. x' =

5 —1 x

3 1 /

—2 1 x

—5 4 /

1 1

0 2

—1 1

0 0

2 0

1 2

x(°) =

x(°) =

2

1

x(°) = I 0

x(°) = I 5

x

x

x

x

x

x

x

382

Chapter 7. Systems of First Order Linear Equations

19. The system tx' = Ax is analogous to thesecond order Eulerequation(Section5.5). Assuming that x = , where ? is a constant vector, showthat ? 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.

(a) Sketch a phase portrait of the system.

(b) Sketch the trajectory passing through the initial point (2, 3).

(c) For the trajectory in part (b) sketch the graphs of x1 versus t and of x2 versus t on the same set of axes.

28. Consider a 2 x 2 system x' = Ax. If we assume that rx = r2, the general solution is

linearly dependent, and then showing that this leads to a contradiction.

(a) Note that ?(1) satisfies the matrix equation (A — ^I)? (1) = 0; similarly, note that

(A — ^I)?(2) = 0.

(b) Show that (A — r2I)?(1) = (r1 — r2)? (1).

(c) Suppose that ?(1) and ?(2) are linearly dependent. Then q ?(1) + c2? (2) = 0 and at least one of c1 and c2 is not zero; suppose that c1 = 0. Showthat (A — r2I)(c1^ (1) + c2?(2)) = 0, and also show that (A — r2I)(c1^(1) + c2? (2)) = c1 (r1 — r2)?(1). Hence c1 = 0, which is

a contradiction. Therefore ?(1) and ? (2) are linearly independent.

(d) Modify the argument of part (c) in case c1 is zero but c2 is not.

(e) Carry out a similar argument for the case in which the order n is equal to 3; note that the procedure can be extended to cover an arbitrary value of n.

29. Consider the equation

where a, b, and c are constants. In Chapter 3 it was shown that the general solution depended on the roots of the characteristic equation

x = Cj?(1) er1t + c2?(2) er2(, provided that ?(1) and ?(2) are linearly independent. In this problem we establish the linear independence of ?(1) and ?(2) by assuming that they are

ay" + b? + cy = 0,

(i)

ar2 + br + c = 0.

(ii)

7.5 Homogeneous Linear Systems with Constant Coefficients

383

(a) Transform Eq. (i) into a system of first order equations by letting x1 = y, x2 = y. Find the system of equations x' = Ax satisfied by x = ^.

(b) Find the equation that determines the eigenvalues of the coefficient matrix A in part (a). Note that this equation is just the characteristic equation (ii) of Eq. (i).

? 30. The two-tank system of Problem 21 in Section 7.1 leads to the initial value problem

1 3

x' =1 1 41 I x, x(0) =

10 5

-17

-21

where x1 and x2 are the deviations of the salt levels Q1 and Q2 from their respective equilibria.

(a) Find the solution of the given initial value problem.

(b) Plot x1 versus t and x2 versus t on the same set of axes.

(c) Find the time T such that |x1 (t) | < 0.5 and |x2(t) | < 0.5 for all t > T.

31. Consider the system

x' = (: :Dx

(a) Solve the system for a = 0.5. What are the eigenvalues of the coefficient matrix? Classify the equilibrium point at the origin as to type.

(b) Solve the system for a = 2. What are the eigenvalues of the coefficient matrix? Classify the equilibrium point at the origin as to type.

(c) In parts (a) and (b) solutions of the system exhibit two quite different types of behavior. Find the eigenvalues of the coefficient matrix in terms of a and determine the value of a between 0.5 and 2 where the transition from one type of behavior to the other occurs. This critical value of a is called a bifurcation point.

Electric Circuits. Problems 32 and 33 are concerned with the electric circuit described by the system of differential equations in Problem 20 of Section 7.1:

y) ? (1)

32. (a) Find the general solution of Eq. (i) if R1 = 1 ohm, R2 = 5 ohm, L = 2 henrys, and C = 2 farad.

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