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

ISBN 0-471-31999-6

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For second order systems with real coefficients we have now completed our description of the three main cases that can occur.

1. Eigenvalues have opposite signs; x = 0 is a saddle point.

390

Chapter 7. Systems of First Order Linear Equations

PROBLEMS

2. Eigenvalues have the same sign but are unequal; x = 0 is a node.

3. Eigenvalues are complex with nonzero real part; x = 0 is a spiral point.

Other possibilities are of less importance and occur as transitions between two of the cases just listed. For example, a zero eigenvalue occurs during the transition between a saddle point and a node. Purely imaginary eigenvalues occur during a transition between asymptotically stable and unstable spiral points. Finally, real and equal eigenvalues appear during the transition between nodes and spiral points.

In each of Problems 1 through 8 express the general solution of the given system of equations in terms of real-valued functions. In each of Problems 1 through 6 also draw a direction field, sketch a few of the trajectories, and describe the behavior of the solutions as t ^<x>.

1. x' = x ? 2. x' =

3. x' = x ? 4. x' = n 2 x

'-1 -4

11

1

5. x' = I _ „ I x ? 6. x' =

' 1 2

>-5 -1

0 0\ 1-3 0

7. x' = | 2 1 -2 I x 8. x' = I 1 -1

2 1/ \-2 -1

In each of Problems 9 and 10 find the solution of the given initial value problem. Describe the behavior of the solution as t .

9. x'=0 -3)x x(0)=0

10. x' = (-3 -1) x x(0) = (-1

In each of Problems 11 and 12:

(a) Find the eigenvalues of the given system.

(b) Choose an initial point (other than the origin) and draw the corresponding trajectory in the x1 x2-plane.

(c) For your trajectory in part (b) draw the graphs of x1 versus t and of x2 versus t.

(d) For your trajectory in part (b) draw the corresponding graph in three-dimensional tx1 x2-space.

nw=(1 :_2) x ? 12 ^=(:; 2)x

In each of Problems 13 through 20 the coefficient matrix contains a parameter a. In each of these problems:

(a) Determine the eigenvalues in terms of a.

(b) Find the critical value or values of a where the qualitative nature of the phase portrait for the system changes.

(c) Draw a phase portrait for a value of a slightly below, and for another value slightly above, each critical value.

x

7.6 Complex Eigenvalues

391

? 13. x' =

? 15. x'

? 17. x'

? 19. x' =

a 1 -1 a

2 -5

a —2

-1 a -1 -1

a 10 -1 -4

? 14. x'

? 16. x' =

? 18. x' =

? 20. x' =

05

64

4a

86

In each of Problems 21 and 22 solve the given system of equations by the method of Problem 19 of Section 7.5. Assume that t > 0.

21. tx =

-1 -1 21

22. =| j -2 , x

In each of Problems 23 and 24:

(a) Find the eigenvalues of the given system.

(b) Choose an initial point (other than the origin) and draw the corresponding trajectory in the Vjx2-plane. Also draw the trajectories in the xxx3- and x2x3-planes.

(c) For the initial point in part (b) draw the corresponding trajectory in x1 x2x3-space.

<-4 1 0 <-4 1 0

? 23. x' = -1 1 4 0 x ? 24. x' = -1 l 4 0 x

0 0 - h 0 0 10>

25. Consider the electric circuit shown in Figure 7.6.5. Suppose that R1 = R2 = 4 ohms, C = 2 farad, and L = 8 henrys.

(a) Show that this circuit is described by the system of differential equations

-I1

dt\V

(1)

where I is the current through the inductor and V is the voltage drop across the capacitor. Hint: See Problem 18 of Section 7.1.

(b) Find the general solution of Eqs. (i) in terms of real-valued functions.

(c) Find I(t) and V(t) if I(0) = 2 amperes and V(0) = 3 volts.

(d) Determine the limiting values of I(t) and V(t) as t -—?. Do these limiting values depend on the initial conditions?

x

x

1

a

x

x

a

3

a

x

x

x

x

x

FIGURE 7.6.5 The circuit in Problem 25.

392

Chapter 7. Systems of First Order Linear Equations

26. The electric circuit shown inFigure 7.6.6 is described by the system of differential equations

dt\V

0

I \ L

1

V_C

1 \ V

RC>

(i)

where I is the current through the inductor and V is the voltage drop across the capacitor. These differential equations were derived in Problem 18 of Section 7.1.

(a) Show that the eigenvalues of the coefficient matrix are real and different if L > 4 R2C; show that they are complex conjugates if L < 4 R2C.

(b) Suppose that R = 1 ohm, C = 1 farad, and L = 1 henry. Find the general solution of the system (i) in this case.

(c) Find I(t) and V(t) if I(0) = 2 amperes and V(0) = 1 volt.

(d) For the circuit of part (b) determine the limiting values of I(t) and V(t) as t ^ro. Do these limiting values depend on the initial conditions?

C

—1(-

R

-Wr

L

FIGURE 7.6.6 The circuit in Problem 26.

27. In this problem we indicate how to show that u(t) and v(t), as given by Eqs. (9), are linearly independent. Let r1 = k + ip and r 1 = k — ip be a pair of conjugate eigenvalues of the coefficient matrix A of Eq. (1); let ^ (1) = a + ib and ^(1) = a — ib be the corresponding eigenvectors. Recall that it was stated in Section 7.3 that if r1 = r1, then ^(1) and ^ (1) are linearly independent.

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