Consider the differential equation dx/dt = kx x3 (a) If k

Problem 1P In problem first solve the equation f(x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f(x). Then analyze the sign of f(x) to determine whether each critical point is stable or unstable, and construct the comsfx/iuling phase diagram for the differential equation. Next, solve the differential equation explicitly for x(t) in terms of t. Finally, use either the exact solution or a computer-generated slope field to sketch typical solution curves for the given differential equation, and verify visually the stability of each critical point.

Problem 2P In problem first solve the equation f(x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f(x). Then analyze the sign of f(x) to determine whether each critical point is stable or unstable, and construct the comsfx/iuling phase diagram for the differential equation. Next, solve the differential equation explicitly for x(t) in terms of t. Finally, use either the exact solution or a computer-generated slope field to sketch typical solution curves for the given differential equation, and verify visually the stability of each critical point.

Problem 4P In problem first solve the equation f(x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f(x). Then analyze the sign of f(x) to determine whether each critical point is stable or unstable, and construct the comsfx/iuling phase diagram for the differential equation. Next, solve the differential equation explicitly for x(t) in terms of t. Finally, use either the exact solution or a computer-generated slope field to sketch typical solution curves for the given differential equation, and verify visually the stability of each critical point.

Problem 5P In problem first solve the equation f(x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f(x). Then analyze the sign of f(x) to determine whether each critical point is stable or unstable, and construct the comsfx/iuling phase diagram for the differential equation. Next, solve the differential equation explicitly for x(t) in terms of t. Finally, use either the exact solution or a computer-generated slope field to sketch typical solution curves for the given differential equation, and verify visually the stability of each critical point.

Problem 29P Consider the two differential equations each having the critical points a, b, and c; suppose that a < b < c. For one of these equations, only the critical point b is stable; for the other equation, b is the only unstable critical point. Construct phase diagrams for the two equations to determine which is which. Without attempting to solve either equation explicitly, make rough sketches of typical solution curves for each. You should see two funnels and a spout in one case, two spouts and a funnel in the other. i FIGURE. Solution curves for harvesting a population of alligators. Then substitute the expressions given in (22) for x? and y? to obtain Tt follows that Then differentiation of r2 = x2 + y2 yields so r = r(t) satisfies the differential equation FIGURE 1. Spiral trajectories of the system in Eq. (22) with Ic = 5. In Problem 22 we ask you to derive the solution where fq = r(0). Thus the typical solution of Eq. (22) may be expressed in the form If r0 = 1, then Eq. (25) gives r(t) = 1 (the unit circle). Otherwise, if r0 > 0, then Eq. (25) implies that r(t) ? > 1 as 0. Hence the trajectory defined in (26) spirals in toward the unit circle if f(q) < 1 and spirals out toward this closed trajectory if 0 > ly > 1. Figure 1 shows a trajectory spiraling outward from the origin and four trajectories spiraling inward, all approaching the closed trajectory r(t) = 1. ? Under rather general hypotheses it can be shown that there are four possibilities for a nondegenerate trajectory of the autonomous system The tour possibilities are these: 1. (x(f), y(t)) approaches a critical point as t + ?. 2. (x(t), y(t)) is unbounded with increasing t. 3. y(t)) is a periodic solution with a closed trajectory. 4. (x(f), y(t)) spirals toward a closed trajectory as t? + ?. As a consequence, the qualitative nature of the phase plane picture of the trajectories of an autonomous system is determined largely by the locations of its critical points and by the behavior of its trajectories near its critical points. We will see in Section 7.3 that, subject to mild restrictions on the functions F and G, each isolated critical point of the system x? = F(x, y) y? = G(x, y) resembles qualitatively one of the examples of this section=it is cither a node (proper or improper), a saddle point, a center, or a spiral point.

Problem 6P In problem first solve the equation f(x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f(x). Then analyze the sign of f(x) to determine whether each critical point is stable or unstable, and construct the comsfx/iuling phase diagram for the differential equation. Next, solve the differential equation explicitly for x(t) in terms of t. Finally, use either the exact solution or a computer-generated slope field to sketch typical solution curves for the given differential equation, and verify visually the stability of each critical point.

Problem 3P In problem first solve the equation f(x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f(x). Then analyze the sign of f(x) to determine whether each critical point is stable or unstable, and construct the comsfx/iuling phase diagram for the differential equation. Next, solve the differential equation explicitly for x(t) in terms of t. Finally, use either the exact solution or a computer-generated slope field to sketch typical solution curves for the given differential equation, and verify visually the stability of each critical point.

Problem 7P In problem first solve the equation f(x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f(x). Then analyze the sign of f(x) to determine whether each critical point is stable or unstable, and construct the comsfx/iuling phase diagram for the differential equation. Next, solve the differential equation explicitly for x(t) in terms of t. Finally, use either the exact solution or a computer-generated slope field to sketch typical solution curves for the given differential equation, and verify visually the stability of each critical point.

Problem 8P In problem first solve the equation f(x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f(x). Then analyze the sign of f(x) to determine whether each critical point is stable or unstable, and construct the comsfx/iuling phase diagram for the differential equation. Next, solve the differential equation explicitly for x(t) in terms of t. Finally, use either the exact solution or a computer-generated slope field to sketch typical solution curves for the given differential equation, and verify visually the stability of each critical point.

Problem 9P In problem first solve the equation f(x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f(x). Then analyze the sign of f(x) to determine whether each critical point is stable or unstable, and construct the comsfx/iuling phase diagram for the differential equation. Next, solve the differential equation explicitly for x(t) in terms of t. Finally, use either the exact solution or a computer-generated slope field to sketch typical solution curves for the given differential equation, and verify visually the stability of each critical point.

Problem 10P In problem first solve the equation f(x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f(x). Then analyze the sign of f(x) to determine whether each critical point is stable or unstable, and construct the comsfx/iuling phase diagram for the differential equation. Next, solve the differential equation explicitly for x(t) in terms of t. Finally, use either the exact solution or a computer-generated slope field to sketch typical solution curves for the given differential equation, and verify visually the stability of each critical point.

Problem 11P In problem first solve the equation f(x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f(x). Then analyze the sign of f(x) to determine whether each critical point is stable or unstable, and construct the comsfx/iuling phase diagram for the differential equation. Next, solve the differential equation explicitly for x(t) in terms of t. Finally, use either the exact solution or a computer-generated slope field to sketch typical solution curves for the given differential equation, and verify visually the stability of each critical point.

Problem 12P In problem first solve the equation f(x) = 0 to find the critical points of the given autonomous differential equation dx/dt = f(x). Then analyze the sign of f(x) to determine whether each critical point is stable or unstable, and construct the comsfx/iuling phase diagram for the differential equation. Next, solve the differential equation explicitly for x(t) in terms of t. Finally, use either the exact solution or a computer-generated slope field to sketch typical solution curves for the given differential equation, and verify visually the stability of each critical point.

Problem 13P In Problem, use a computer system or graphing calculator to plot a slope field and/or enough solution caws to indicate the stability or instability of each critical point of the given differential equation. (Some of these critical points may he semistable in the sense mentioned in Example.) \

Problem 14P In Problem, use a computer system or graphing calculator to plot a slope field and/or enough solution caws to indicate the stability or instability of each critical point of the given differential equation. (Some of these critical points may he semistable in the sense mentioned in Example.)

Problem 15P In Problem, use a computer system or graphing calculator to plot a slope field and/or enough solution caws to indicate the stability or instability of each critical point of the given differential equation. (Some of these critical points may he semistable in the sense mentioned in Example.)

Problem 16P In Problem, use a computer system or graphing calculator to plot a slope field and/or enough solution caws to indicate the stability or instability of each critical point of the given differential equation. (Some of these critical points may he semistable in the sense mentioned in Example.)

Problem 17P In Problem, use a computer system or graphing calculator to plot a slope field and/or enough solution caws to indicate the stability or instability of each critical point of the given differential equation. (Some of these critical points may he semistable in the sense mentioned in Example.)

Problem 18P In Problem, use a computer system or graphing calculator to plot a slope field and/or enough solution caws to indicate the stability or instability of each critical point of the given differential equation. (Some of these critical points may he semistable in the sense mentioned in Example.)

Problem 19P The differential equation models a logistic population with harvesting at rate h. Determine (as in Example 6) the dependence of the number of critical points on the parameter h, and then construct a bifurcation diagram like Fig. 2.2.12

Problem 20P The differential equation models a population with stocking at rate s. Determine the dependence of the number of critical points c on the parameter s and then construct the corresponding bifurcation diagram in the sc-plane.

Problem 21P Consider the differential equation dx/dt = kx ? x3 (a) If k 0, show that the only critical value c ? 0 of x is stable. (b) If k > 0, show that the critical point c = 0 is now unstable, but that the critical points are stable. Thus the qualitative nature of the solutions changes at k = 0 as the parameter k increases, and so k = 0 is a bifurcation point for the differential equation with parameter k. The plot of all points of the form (k, c) where c is a critical point of the equation x? = kx- x3 , is the “pitchfork diagram” shown in Fig. FIGURE. Bifurcation diagram for dx/dt = kx ? x3.

Problem 22P Consider the differential equation dx/dt = x + kx3 containing the parameter k. Analyze (as in Problem) the dependence of the number and nature of the critical points on the value of k, and construct the corresponding bifurcation diagram. Problem Consider the differential equation dx/dt = kx ? A3 (a) If k > 0, show that the only critical value c ? 0 of x is stable. (b) If k > 0. show that the critical point c = 0 is now unstable, but that the critical points stable. Thus the qualitative nature of the solutions changes at k ? 0 as the parameter k increases, and so k = 0 is a bifurcation point for the differential equation with parameter k. The plot of all points of the form (k, c) where c is a critical point of the equation x? = kx- A"3 is the “pitchfork diagram” shown in Fig. FIGURE. Bifurcation diagram for dx/dt = kx ? x3.

Problem 26P Example 4 dealt with the case 4h > kM2 in the equation dx/dt = kx(M ? x) = h that describes constant-rate harvesting of a logistic population. Problems deal with the other cases. If 4h = kM2, show that typical solution curves look as illustrated in Fig. Thus if x0 M/2, then then x(t) = 0 after a finite period of time, so the lake is fished out. The critical point x = M/2 might be called semistable, because it looks stable from one side, unstable from the other.

Problem 27P Example 4 dealt with the case Ah > kM2 in the equation dx/dt = kx(M ? x) = h that describes constant-rate harvesting of a logistic population. Problems deal with the other cases. If 4h > kM2, show that x(r) = 0 after a finite period of time, so the lake is fished out (whatever the initial population). [Suggestion: Complete the square to rewrite the differential equation in the form dx/dt = ?k\(x ? a)2 + b2]. Then solve explicitly by separation of variables.] The results of this and the previous problem (together with Example 4) show that is a critical harvesting rate for a logistic population. At any lesser harvesting rate the population approaches a limiting population N that is less than M(why?), whereas at any greater harvesting rale the population reaches extinction. FIGURE. Solution curves for harvesting a logistic population with 4h ? kM1.

Problem 28P This problem deals with the differential equation dx/dt = kx(x ? M)-h that models the harvesting of an unsophisticated population (such as alligators). Show that this equation can be rewritten in the form dx/dt = k(x ? H)(x ? K), where Show that typical solution curves look as illustrated in Figure 2.2.15.

Problem 23P Suppose that the logistic equation dx/dt = kx(M ? x) models a population x(t) of fish in a lake after t months during which no fishing occurs. Now suppose that, because of fishing, fish are removed from the lake at the rate of lix fish per month (with h a positive constant). Thus fish are “harvested” at a rate proportional to the existing fish population, rather than at the constant rate of Example. (a) II 0 > h < kM, show that the population is Still logistic. What is the new limiting population? If show that so the lake is eventually fished out.

Problem 24P Separate variables in the logistic harvesting equation dx/dt = k(N ? x)(x ? H) and then use partial fractions to derive the solution given in Eq.

Problem 25P Use the alternative forms of the solution in to establish the conclusions staled.