MAT244 - Ordinary Differential Equations - Summer 2016 Assignment 1 Due: June 8, 2016 Full Name: Last First Student #: Indicate which Tutorial Section you attend by filling in the appropriate circle: Tut 01 M 2-3 pm BA1240 Christopher Adkins Tut 02 M 3-4 pm BA1240 Fang Shalev Housfater Tut 05 W 5-6 pm BA2165 Krishan Rajaratnam Print out this page, fill it out and attach it to the front of your assignment. Instructions: Due June 8, 2016 at the start of the lecture 13:10pm in BA1170. You may collaborate with your classmates but you MUST write up your solutions independently. Write your solutions clearly, showing all steps. Do not submit just your rough work. Grading is based on both the correctness and thee presentation of your answer. Late assignments will not be accepted without appropriate documentation to explain the lateness (eg. a UofT medical note) Assignments may be submitted to the course instructor for remarking up to one week after they are returned. If you request a regrade, make sure your assignment is written in pen and attach a note explaining clearly which part and why you believe it was graded incorrectly. Note that it is possible for grades to be reduced after remarking . For grader use: Q1 Q2 Q3 Q4 Q5 Total Answer the following questions. Each is worth 20 points for a total of 100. 1. The Linear Problem Consider then linear differential equation for y(t) y 0 (t) t y=t 1 t2 (a) Find the general solution to this equation (using integrating factors is helpful) (b) Find the particular solution corresponding to the initial condition y(t = 0) = 2 (c) On which interval of values of t is the solution in part b) valid? 2. The Fishery Problem Suppose that you operate a salmon farm where your fish reproduce at a natural rate with the logistic model with rate r and optimal population P . Without external factors the population P obeys the standard equation : \u0012 \u0013 P dP =r 1 P dt P . This being a farm, you harvest the fish at a rate A proportional to its population. So 0 < A < r (you don't want to eat the fish faster than they can reproduce). Thus the equation for the population becomes 1 (a) Show that PEQ dP = r (1 P/P ) P AP dt \u0001 2 1 = 0 and PEQ = P 1 Ar are equilibrium points. Show that PEQ is 2 unstable while PEQ is stable. Include a diagram with a few solution curves. (b) Suppose you want to maximize the yield of fish you eat while keeping the population 2 stable near the value PEQ . What value of the catching rate A optimizes the yield Y = AN ? What percentage of fish population do you eat per year at the optimal rate? 3. The Red Tide attacks the fishery Now an unfortunate event happens and a bloom of algae called a red tide https://en.wikipedia.org/wiki/Red_tide poisons a number of your fish. The algae poison your fish at a rate R > 0 independent of the current population. The equation for the fish population is now \u0012 \u0013 P dP =r 1 P R AP dt P (a) Show that there is a critical rate of poisoning Rcrit = P 4r (r A)2 so that if R < Rcrit 1 2 1 2 there are still equilibrium solutions PEQ < PEQ such that PEQ is unstable while PEQ is stable. Include a diagram that shows a few solution curves. 1 (b) Show that PEQ > 0. Suppose that you initially had a small number of fish P (t = 0) = 1 P0 < PEQ . Prove that all your fish die in finite time. (A diagram might help) (c) Typically the food demands remain the same and people continue to fish at the same rate they did in normal circumstance. Suppose you continue to catch fish at the rate you obtained in problem 2)b). Prove that if the poisoning rate gets too strong, namely R > P r , 16 all your fish die in finite time no matter how many you had at the start. (Again a diagram might help) Incidentally, the fishermen protesting and blocking roads due to an ongoing event of red tide in southern Chile is the reason we started the course a lecture late. 4. The Exact Problem (a) Solve the following equation for y(t). Leave the solution in implicit form: dy 2 yety = dt 2y + tety (b) Check if the following equation for y(x) is exact: \u0001 dy = 0. dx If it is not exact, find an integrating factor that makes it exact and solve for y(x). You y + 3xy e3y can leave the solution in implicit form. (hint: multiply both sides of the equation by an appropriate function (y).) 5. The Uniqueness Problem (Don't be scared by the many parts, only e) is challenging.) Consider the nonlinear differential equation y 0 (x) = x arcsin(y) (1) Note: you may have seen arcsin(y) denoted as sin1 (y) before. It is simply the inverse function to the sine restricted to [ 2 , 2 ]. (a) Show that given the initial condition y(x = 0) = y0 = 0, then y(x) = 0 is a solution to (1) on its domain (b) Suppose that there is another solution y2 (x) to (1) with y2 (x = 0) = 0 but y2 (x) 6= 0 for some other value of x. Because y2 is continuous, show there is some s > 0 so that |y2 (x)| < 1 2 whenever x [s, s]. (c) Using the mean value theorem (and the fact that |y2 | < arcsin(y2 (x)) 2 y2 (x) 3 1 2 for these x values!), show that for any x [s, s] (d) Applying the fundamental theorem of calculus, show that for any x [s, s] Z x x arcsin(y2 ( x))d x Z x 2s |y2 ( x)|d x 3 0 y2 (x) = 0 (e) Let F (x) = Rx 0 |y2 ( x)|d x. By part d), we have that F 0 (x) 2s F (x). 3 Using that F (0) = 0, show F (x) = 0 for every x [s, s] and thus y2 (x) = 0 as well by continuity (hint: Look up Gronwall's lemma). Therefore the solution y(x) = 0 for every x is unique near to 0! (f) Why will this argument fail if I had said y(x = 0) = y0 = 1 in part a)