15 Sofsomee relation8a. Fromcarryingorrespondversusside ofe obtain(18GGGProblemsProblems 1 through 4 involve equations of the form dy/dt = f(y). Ineach problem sketch the graph of f(y) versus y, determine the critical(equilibrium) points, and classify each one as asymptotically stable orunstable. Draw the phase line, and sketch several graphs of solutionsin the ty-plane.G 1.dy/dt = ay+by²2, a > 0, b> 0,dy/dt = y(y - 1)(y-2), Yo ≥ 02.3.dy/dt = e' -1, -∞< Yo <∞G4. dy/dt = e- - 1,-00< Yo <∞5. Semistable Equilibrium Solutions. Sometimes a constantequilibrium solution has the property that solutions lying on one sideof the equilibrium solution tend to approach it, whereas solutions lyingon the other side depart from it (see Figure 2.5.9). In this case theequilibrium solution is said to be semistable.a. Consider the equationStabscontkGGу4dy/dt = k(1- y)²,(19)where k is a positive constant. Show that y = 1 is the only criticalpoint, with the corresponding equilibrium solution (t) = 1.Gb. Sketch f(y) versus y. Show that y is increasing as afunction of t for y< 1 and also for y> 1. The phase linehas upward-pointing arrows both below and above y = 1. Thussolutions below the equilibrium solution approach it, and thoseabove it grow farther away. Therefore, (t) = 1 is semistable.c. Solve equation (19) subject to the initial condition y(0) = yoand confirm the conclusions reached in part b.o(t) = ktYA-∞o Yo <∞ko(t) = kProblems 6 through 9 involve equations of the form dy/dt = f(y).In each problem sketch the graph of f(y) versus y, determine thecritical (equilibrium) points, and classify each one as asymptoticallystable, unstable, or semistable (see Problem 5). Draw the phase line,and sketch several graphs of solutions in the ty-plane.G6.dy/dt = y²(y²-1),-∞ T. That is, derive formula (16) byfinding the value of when the denominator of the right-hand side ofequation (15) is zero.13. Complete the derivation of formula (18) for the locations of theinflection points of the solution of the logistic growth model with athreshold (17). Hint: Follow the steps outlined on p. 66.14. Consider the equation dy/dt = f(y) andsuppose that yi is acritical point that is, f(y₁) = 0. Show that the constant equilibriumsolution (t) = y₁ is asymptotically stable if f'(y₁) < 0 and unstableif f'(y₁) > 0. bool to contos ads siling of sids15. Suppose that a certain population obeys the logistic equationdy/dt = ry(1-(y/K)).a. If yo = K/3, find the time at which the initial populationhas doubled. Find the value of 7 corresponding to r = 0.025 perSur year.mueen of kin191Ab. If yo/K = a, find the time T at which y(T)/K = ß,where 0

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(19) y critical = 1. ing as a mase line 1. Thus and those istable. 0) = yo if f'(y₁) > 0. bol to 15. Suppose that a certain population obeys the logistic equation dy/dt = ry(1-(y/K)). = a. If yo K/3, find the time T at which the initial population has doubled. Find the value of 7 corresponding to r = = 0.025 per year. Tab. If yo/K = a, find the time T at which y(T)/K = 3, where 0 → 0 or a, 3 < 1. Observe that T→ ∞o as a nas 3- →1. Find the value of T for r = 0.025 per year, a = 0.1, and 3 = 0.9. G 16. Another equation that has been used to model population Shr