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

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9. x'=0 _5)x’ x(0)=G

10. x'=(_3 _1)x x(0) = \ -2

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.

"¦*=(! _)x > i2-x4_ 2

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' =

à 1 -1 à

2 -5

à2

-1 à -1 -1

à 10 -1 -4

> 14. x' =

> 16. x' =

> 18. x' =

> 20. x' =

05

64

4à

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. =l2 -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 x1x2-plane. Also draw the trajectories in the x1x3- 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 0 x > 24. x' = -1 1 0 x

4 4

0 0 - Ü 0 0

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

-(I

dt I V

(i)

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 -—ro. Do these limiting values depend on the initial conditions?

x

x

1

à

x

x

à

3

à

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

\~C

1 V 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 ^æ. Do these limiting values depend on the initial conditions?

C

4(-

R

-ËËËã-

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 = X + ip and r 1 = X — ip be a pair of conjugate eigenvalues of the coefficient matrix A of Eq. (1); let g (1) = a + ib and g(1) = a — ib be the corresponding eigenvectors. Recall that it was stated in Section 7.3 that if r1 = r1, then g(1) and g (1) are linearly independent.

(a) First we show that a and b are linearly independent. Consider the equation c1a + c2b = 0. Express a and b in terms of g(1) and g (1), and then show that (c1 — ic2)g(1) + (q + ic2)g (1) = 0.

(b) Show that c1 — ic2 = 0 and c1 + ic2 = 0 and then that c1 = 0 and c2 = 0. Conse-

quently, a and b are linearly independent.

(c) To show that u(t) and v(t) are linearly independent consider the equation c1u(t0) + c2v(t0) = 0, where t0 is an arbitrary point. Rewrite this equation in terms of a and b, and then proceed as in part (b) to show that c1 = 0 and c2 = 0. Hence u(t) and v(t) are linearly independent at the arbitrary point t0. Therefore they are linearly independent at every point and on every interval.

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