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

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which is either a node or spiral point (depending on how

the roots bifurcate) which is unstable.

6a. The critical points are the solutions of x(1-x-y) = 0 and y(3-x-2y) = 0. Solutions are x = 0, y = 0; x = 0,

3 - 2y = 0 which gives y = 3/2; y = 0 and 1 - x = 0 which

give x = 1; and 1 - x - y = 0, 3 - x - 2y = 0 which give

x = -1, y = 2. Thus the critical points are (0,0), (0,3/2), (1,0) and (-1,2).

6b, For the critical point (0,0) the D.E. is already in the

6c. form of an almost linear system; and the corresponding linear system is du/dt = u, dv/dt = 3v which has the eigenvalues r1 = 1 and r2 = 3. Thus the critical point (0,0) is an unstable node. Each of the other three critical points is dealt with in the same manner; we consider only the critical point (-1,2). In order to translate this critical point to the origin we set x(t) = -1 + u(t), y(t) = 2 + v(t) and substitute in the D.E. to obtain

du/dt = -1 + u - (-1+u)2 - (-1+u)(2+v) = u + v - u2 - uv

and

dv/dt = 3(2+v) - (-1+u)(2+v) - 2(2+v)2 = -2u - 4v - uv - 2v2.

Writing this in the form of Eq.(4) we find that

A =

and g =

2

vuv + 2v ó

which is an almost

linear system. The eigenvalues of the corresponding linear system are r = (-3 ± ä/"9 + 8 )/2 and hence the critical point (-1,2), of the original system, is an unstable saddle point.

22

10a. The critical points are solutions of x + x + y = 0 and y(1-x) = 0, which yield (0,0) and (-1,0).

10b. For (0,0) the D.E. is already in the form of an almost

linear system and thus du/dt = u and dv/dt = v. For

(-1,0) we let u = x+1, v = y so that substituting x = u-1

du 2 2

and y = v into the D.E. we obtain — = -u + u + v and

dt

v1y

2

u

+ uv

184

Section 9.3

dv

dt

u'

= 2v - uv. Thus the corresponding linear system is = -u and v 10c. For (0,0) A =

= -2v.

1 ox

r = 1, so that

2

which has r v0 1) 1

(0,0), for the nonlinear system, will be either a node or spiral point, depending on how the roots bifurcate. In any case, since r and r are positive, the system will

be unstable. For (-1,0) A

/ _ _ \ -1 0

and thus

r = -1 and r = 2, and hence the nonlinear system, from Table 9.3.1, has an unstable saddle point at (-1,0).

d ´ \ Ã1 0 ] ¥ \ V 0 1

x x

18a. The system is - = + 3

dt I y) V 0 -2 , I yj

and thus is

almost linear using the procedures outlined in the earlier problems. The corresponding linear system has the eigenvalues r1 = 1, r2 = -2 and thus (0,0) is an unstable saddle point for both the linear and almost linear systems.

18b. The trajectories of the linear system are the solutions of dx/dt = x and dy/dt = -2y and thus x(t) = c1et and y(t) = c2e-2t. To sketch these, solve the first equation for et and substitute into the second to obtain Ó = c2c2/x2, Ô 0.

Several trajectories are shown in the figure.

Since x(t) = c1et, we must pick c1 = 0 for x ^ 0 and t ^ •. Thus x = 0, y = c2e-2t (the vertical axis) is the only trajectory for

M

(W

18c

which x ^ 0, y ^ 0 as t ^ •.

For x Ï 0 we have dy/dx = (dy/dt)/(dx/dt) = (-2y+x3)/x.

This is a linear equation, and the general solution is y = x3/5 + k/x2, where k is an arbitrary constant. In addition the system of equations has the solution x = 0, y = Be-2t. Any solution with its initial point on the y-axis (x=0) is given by the latter solution. The trajectories corresponding to these solutions approach

Section 9.3

185

the origin as t —> oo. The trajectory that passes through the origin and divides the family of curves is given by k = 0, namely y = x3/5. This trajectory corresponds to the trajectory y = 0 for the linear problem.

Several trajectories

are sketched in the figure

22a.

2 Ï

22b. From the graphs in part a, we see that vc is between v = 2

and v = 5. Using several values for v, we estimate vc @ 4.00.

23a.

For v = 2, the motion is damped oscillatory about x = 0. For v = 5, the pendulum swings all the way around once and then is a damped oscillation about x = 2p (after one full rotation). For problem 22, this later case is not damped, so x continues to increase, as shown earlier.

27a. Setting c = 0 in Eq.(10) of Section 9.2 we obtain

mL2d20/dt2 + mgLsin0 = 0. Considering d0/dt as a function of 0 and using the chain rule we have

d ( d0 ) d ( d0 ) d0 1 d ( ) = ( ) = dt dt d0 dt dt 2

(1/2)mL2d[(d0/dt) 2]/d0 = -mgLsin0. Now integrate both sides from a to 0 where d0/dt = 0 at 0 = a:

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