APM1514 Assignment 5 2024 - DUE 27 June 2024

APM1514 Assignment 5 2024 - DUE 27 June 2024 ;100 % TRUSTED workings, explanations and solutions. For assistance call or W.h.a.t.s.a.p.p us on ...(.+.2.5.4.7.7.9.5.4.0.1.3.2)........... Question 1: 14 Marks (1.1) By using an analytical approach, determine all the equilibrium point(s) of the following dif- (14) ferential equation: dx dt = 3 ⇣p x 3 1 ⌘ 1 2 ⇣p x 3 1 ⌘ 1 65. Question 2: 24 Marks Draw the phase lines of the following differential equations. List all the equilibrium points for each system, and classify them as stable or unstable: (2.1) (8) dx dt = 22x+1 9(2x ) + 4, (2.2) (8) dx dt = 32x+1 28(3x ) + 9, (2.3) (8) dx dt = x ln x x 3 e 2 . Question 3: 27 Marks For each of the following systems, (3.1) dx dt = 8x , (3.2) dx dt = (x 2 8x ), 32 APM1514/101/0/2024 (3.3) dx dt = (x 2 4x + 4)(x 2 + x ), Draw the sketch of: (a) dx dt as a function of x, (b) the phase line, (c) also, for each system, list all the equilibrium points and classify each of them as stable or unstable, (d) and predict the outcome of the solution if the system starts at x (0) = 0.5. Question 4: 20 Marks Consider the model dP dt = 5P + 10P 2 where P(t )represents the size of the population at time t. (4.1) Is this a logistic model? Justify your answer! (4.2) Draw a phase line of the model. (4.3) Draw a sketch of the solution curve P(t ), if P(0) = 5. Question 5: 15 Marks Assume that for the system dx dt = G(x) a graph of the function G looks like this: Draw the phase line of the system, and explain what will happen to x (t )as t increases if the initial value is 1 and if the initial value is 3?