MAT 127: Calculus C, Spring 2017 Homework Assignment 6 WebAssign Problems due before 9am, Wednesday, 03/08 20% bonus for submissions before 9am, Saturday, 03/04 Written Assignment due before 12noon, Thursday, 03/09 in your instructor's office Please read Sections 7.6 and the first half 8.1 in the textbook thoroughly before starting on the corresponding problems below. Written Assignment: 7.6 2,6,8; 8.1 22,33; Problems F (below) Briefly explain your answer on all textbook problems and show your work on the letter problem Please write your solutions legibly; the graders may disregard solutions that are not readily readable. All solutions must be stapled (no paper clips) and have your name (first name first), lecture number (L01, L02, L03, or L04), and HW number in the upper-right corner of the first page. Problem F According to the book, the solutions (x, y) = (x(t), y(t)) to the system of differential equations ( dx dt = ax bxy (1) dy dt = cy + dxy with certain constants a, b, c, d > 0 trace simple closed curves (loops) in the xy-plane. Let's see why. (a) Divide the second equation in (1) by the first and solve the resulting equation obtaining y = y(x) implicitly; in doing so assume that x, y > 0 (so only the first quadrant is considered). (b) Fix the constant C in your general solution (this gives a specific solution of the equation in (a)). Show that the values of x, y > 0 that satisfy the equation lie in the interval [mC , MC ] for some mC , MC > 0. Furthermore, for each fixed x > 0 at most two values of y > 0 satisfy the equation; for each fixed y > 0 at most two values of x > 0 satisfy the equation. Hint: Your general solution in (a) should be of the form G(y) = CF (x). Show that F = F (x) and G = G(y) have precisely one critical point in the interval (0, ), which is a minimum in one case and a maximum in the other case. Sketch their graphs with x and y both on the horizontal axis. When do they values in common? (c) Assuming x, y > 0, show that (x (t), y (t)) = 0 if and only if (x(t), y(t)) = (c/d, a/b). (d) Show that every phase trajectory of (1) in the first quadrant of the xy-plane other than the equilibrium point (c/d, a/b) repeatedly traces a closed curve enclosing (c/d, a/b) in the counter-clockwise direction