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

ISBN 0-471-31999-6

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9. Setting x = rcos0, y = rsin0 and using the techniques of

Problem 8 the equations transform to dr/dt = r2 - 2,

d0/dt = -1. This system has a periodic solution r = \p2 ,

0 = -t + t0. If r < \p2 , then dr/dt < 0, and the direction of motion along a trajectory is inward. If r > , then dr/dt > 0, and the direction of motion is

outward. Thus the periodic solution r = \p2 , 0 = -t + t0 is unstable.

11. If F(x,y) = x+y+x3-y2, G(x,y) = -x+2y+x2y+y3/3, then

Fx(x,y) + Gy(x,y) = 1+3x2+2+x2+y2 = 3+4x2+y2. Since the

conditions of Theorem 9.7.2 are satisfied for all x and y, and since Fx + Gy > 0 for all x and y, it follows that the system has no periodic nonconstant solution.

13. Since x = f(t), y = y(t) is a solution of Eqs.(15), we have df/dt = F[f (t),y (t)], dy/dt = G[f (t),y (t)]. Hence on the curve C,

F(x,y)dy - G(x,y)dx = f'(t)y'(t)dt -y/(t) f'(t)dt = 0. It follows that the line integral around C is zero.

However, if Fx + Gy has the same sign throughout D, then the double integral cannot be zero. This gives a contradiction. Thus either the solution of Eqs.(15) is not periodic or if it is, it cannot lie entirely in D.

16a. Setting x' = 0 and solving for y yields y = x3/3 - x + k. Substituting this into y'= 0 then gives

x + .8(x3/3 - x + k) - .7 = 0 Using an equation solver we obtain x = 1.1994, y = -.62426 for k = 0 and x = .80485, y = -.13106 for k = .5. To determine the type of critical points these are, we use Eq.(13) of Section 9.3 to find the

Section 9.8

199

3(1-xC) 3

linear coefficient matrix to be A -

?1/3 -.8/3,

is the critical point. For xc = 1.1994 we obtain complex conjugate eigenvalues with a negative real part, and therefore k = 0 yields an asymptotically stable spiral point. For xc = .80485 the eigenvalues are also complex conjugates, but with positive real parts, so k = .5 yields an unstable spiral point.

16b. Letting k = .1, .2, .3, .4 in the cubic equation of part (a)

and finding the corresponding eigenvalues from the matrix in part (a), we find that the real part of the eigenvalues change sign between k = .3 and k = .4. Continuing to iterate in this fashion we find that for k = .3464 that the real part of the eigenvalue is -.0002 while for k = .3465 the real part is .00005, which indicates k = .3465 is the

critical point for which the system changes from stable to unstable.

16d. You must plot the solution for values of k slightly less than k0, found in part (c), to determine whether a limit cycle exists.

Section 9.8, Page 538

1a. From Eq (6), 1 = -8/3 is clearly one eigenvalue and the other two may be found from 12 + Ill - 10(r-1) = 0 using the quadratic formula.

Ë×1 > ( 0 ¯

X2 II 0

,v^3 7 V 0,

, which requires Xi = X2 = 0

lb. For 1 = 1i we have -10+8/3 -10 0

r -1+8/3 0

v0 0 0

and X3 arbitrary and thus ?(1) = (0,0,1)T.

For 1 = 13 = (-11 + a)/2, where a = Y 81+40r , we have ' 10+(11-a)/2 10 0 Y X1

r -1+(11-a)/2 0 X2

0 0 -8/3+(11-a)/2y^X3 ,

The last line implies X3 = 0 and multiplying the first line by

0

200

Section 9.8

(-9+a)/2 we obtain

(81-a2)/4 10(-9+a)/2

r (9-a)/2

^ V o 1

v^2 J II v o ,

2

Substituting a = 81+40r we have

- 10r -10(9-a)/2

r (9-a)/2

ft > / „ \

V x1' II o

U2 J v o ,

9 - j/ 81+40r . Thus X<3> = -2r

v 0

which is proportional to the answer given in the text. Similar calculations give X<2> •

1c. Simply substitute r = 28 into the answers in parts (a) and (b).

2a. The calculations are somewhat simplified if you let x = b + u, y = b + v, and z = <r-1)+w, where b = V8(r-1)/3 . An alternate approach is to extend Eq.<13> of Section 9.3, which is:

f \

u v

: vwy <xo,yo,zo)

In this example F = -10x + 10y, G = rx H = -8z/3 + xy and thus

\ u ' Fx Fy Fz "

vJ = vv Gx Gy Gz

w V Hx Hy Hzy

/ \ u / o *—i i 1o o > / u

v vJ = r -1 1 * o vv

V wV v yo xo -8/3 j v w

xo = yo = V8<r-1)/3 .

2b. Eq.<9) is found by evaluating

1

b

- y - xz and

q.<8) for P2 since

1o o

-1-1 -b = o.

b -8/3-1

2c. If r = 28, then Eq.<9) is 313 + 4112 + 3o4l + 432o = o, which has the roots -13.8546 and .o93956 ± io.1945i.

3b. If r1,r2,r3 are the three roots of a cubic polynomial, then the polynomial can be factored as

<x-r1)<x-r2)<x-r3). Expanding this and equating to the given polynomial we have A = -<r1+r2+r3),

B = r1r2 + r1r3 + r2r3 and C = -r1r2r3. We are interested in the case when the real part of the complex conjugate roots changes sign. Thus let r2 = a+ib and r3 = a-ib, which yields

A = -<r1+2a), B = 2ar1 + a2 + b2 and C = -r1<a2+b2).

Hence, if AB = C, we have

Section 9.8

201

3c.

4.

5a.

V = 5b.

-(r1+2a)(2ar1+a2+b2) = -r1(a2+b2) or

-2a[r1 + 2ar1 + (a2+b2)] = 0 or -2a[(r1+a)2+b2] = 0.

Since the square bracket term is positive, we conclude that if AB = C, then a = 0. That is, the conjugate complex roots are pure imaginary. Note that the converse is also true. That is, if the conjugate complex roots are pure imaginary then AB = C.

Comparing Eq.(9) to that of part b, we have A = 41/3,

B = 8(r+10)/3 and C = 160(r-1)/3. Thus AB = C yields r = 470/19.

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