MATH 231 WRITTEN HOMEWORK 1 Due at the beginning of class Friday, 9/2/2022 Remember that the purpose of the written homework is for you to practice your technical communication skills, 30 you must show your work and explain your reasoning for full credit. 1. For each initial value problem given below, determine: (i) Whether or not Picard's Existence and Uniqueness Theorem guarantees that a solution exists to the problem. (ii) Whether or not Picard's Existence and Uniqueness Theorem guarantees that a unique solution exists to the problem. Regardless of your conclusion in (i) and (ii) for each problem1 you do not need to solve any of the differential equations. (a) yfg, y(l)=0 (b) $3: + cos(y) : sin{:r) , Ed\") I 0 (c):ii::3x*/$Tla yl2):1 2. When the logistic model is applied to the alligator population, p, on the grounds of Kennedy Space Center in Florida, the following dierential equation is derived: nip , p(p * 1500) E _ 3200 where t is measured in years. (a) Use Maple to draw a direction eld for this DE. Include on the plot some solution curves based on a variety of initial conditions (say, {0, 0), (0, 200), (0, 400), . . . , (0, 1600)). Draw a phase line (by hand) for this situation. Identify the equilibrium solutions and classify them according to their stability. If hunters were allowed to thin the population at a rate of s alligators per year, the equation would be modied to @ _ _p(p 1500) _ alt 3200 (b) Use Maple to draw a direction eld for this di'erential equation when a : 100. Include on the plot some solution curves based on the same initial conditions as part (a). Draw an approximate phase line (by hand) for this situation. You do not have to solve for the exact values of the equilibrium solutions, but identify their approximate locations on your phase line, and classify them according to their stability. 3 1 (c) Use Maple to draw a direction eld for this diiferential equation when a = 200. Include on the plot some solution curves based on the same initial conditions as part (a). Draw a phase line (by hand) for this situation. (d) What do you notice when you compare your three phase lines? What e'ect does the hunters' depletion rate seem to have on the alligator population? Note: Recall that the computers in the Math Center next door to our classroom have Maple installed on them, so you can use them both to load the Maple worksheet from class and to create your own. You should print your nal Maple worksheet, showing both your code and your plots, then you can draw your phase lines next to each respective plot