18.03 Problem Set 1 I encourage collaboration on homework in this course. However, if you do your homework in a group, be sure it works to your advantage rather than against you. Good grades for homework you have not thought through translate to poor grades on exams. You must turn in your own writeups of all problems, and, if you do collaborate, you must write on the front of your solut io n sheet the names of the students you worked with. Because the solutions will be available immediately after the problem sets are due, no ex-tensions will be possible. Part I of each problem set will consist of problems which are either rather routine or for which solutions are available in the back of the book, or in 18.03 Notes and Exercises. Part II contains more challenging and novel problems. They will be graded with care (Com-plain if they are not!) and contribute the bulk of the Homework grade. They will help you develop an understanding of the material. Problems in both parts are keyed closely to the lectures, and numbered to match them. Try the problems as soon as you can after the indicated lecture. Most problem sets corresp ond to four lectures, through the Monday or Wednesday before the set is due. Each problem set is graded out of 100 points. Each day counts 24 points. The grader has 4 points to give or withhold based on neatness and clarity of the answers. (Problem sets 6 and 7 relate to only three lectures, and each day in them counts 32 p oints.) I. First-order diffe re ntial equations R1 T 2 Feb Natural growth models and separable equations: EP 1.1, 1.4. L1 W 3 Feb Direction fields, existence and uniqueness of solutions: EP 1.2, 1.3; Notes G.1; SN 1. R2 Th 4 Feb Isoclines, separatrix, extrema of solutions: Notes G.1, D. L2 F 5 Feb Numerical metho ds : EP 6.1, 6.2; Notes G.2. L3 M 8 Feb Linear equations: models: EP 1.5; SN 2. R3 T 9 Feb Mixing problems, half-life. L4 W 10 Feb Solution of linear equations, integrating factors: EP 1.5; SN 3. Part I. 0. (T 2 Feb) [Natural growth, separable equations] Notes 1A-5c; EP 1.1: 32, 33, 35; EP 1.4: 39, 66; EP 1.5: 1, 9, 20. 1. (W 3 Feb) [Direction fields, isoclines] Notes 1C-1abe. 2. (F 5 Feb) [Euler’s method] Notes 1C-4. 3. (M 8 Feb) [Linear models] EP 1.5: 33, 45. (In both, be sure to write out the differential equation.)� Part II. 0. (T 2 Feb) [Natural growth, separable equations] In recitation a population model was studied in which the natural growth rate of the population of oryx was a constant k > 0, so that for small time intervals Δt the population change x(t + Δt) − x(t) is well approximated by kx(t)Δt. (You also studied the effect of hunting them, but in this problem we will leave that aside.) Measure time in years and the population in kilo-oryx (ko). A mysterious virus infects the oryxes of the Tana River area in Kenya, which causes the growth rate to decrease as time goes on according to the formula k(t) = k0/(a+ t)2 for t ≥ 0, where a and k0 are certain positive constants. (a) What are the units of the constant a in “a + t,” and of the constant k0? (b) Write down the differential e quation modeling this situation. (c) Write down the general solution to your differential equation. Don’t restrict yourself to the values of t and of x that are relevant to the oryx problem; take care of all values of these dx variables. Points to be careful about: use absolute values in x = ln |x| + c correctly, and don’t forget about any “lost” solutions. (d) Now suppose that at t = 0 there is a positive population x0 of oryx. Does the progressive decline in growth rate cause the population stabilize for large time, or does it grow without bound? If it does stabilize, what is the limiting population as t → ∞? 1. (W 3 Feb) [Direction fields, isoclines] In this problem you will study solutions of the differential equation dy 2 dx = y − x . Solutions of this equation do not admit expressions in terms of the standard functions of calculus, but we can study them anyway using the direction field. (a) Draw a large pair of axes and mark off units from −4 to +4 on both. Sketch the direction field given by our equation. Do this by first sketching the isoclines for slopes m = −1, m = 0, m = 1, and m = 2. On this same graph, s ketch, as best you can, a couple of solutions, using just the information given by these four isoclines. Having done this, you will continue to investigate this equation using one of the Mathlets. So invoke http://math.mit.edu/mathlets/mathlets in a web browser and select Isoclines from the menu. (To run the applet from this window, click the little black box with a white triangle inside.) Play around with this applet for a little while. The Mathlets have many features in common, and once you get used to one it will be quicker to learn how to operate the next one. Clicking on “Help” pops up a window with a brief description of the applet’s functionalities. Select from the pull-down menu our differential equation y� = y2 − x. Move the m slider to m = −2 and release it; the m = −2 isocline is drawn. Do the same for m = 0, m = 1, and m = 2. Compare with your sketches. Then depress the mousekey over the graphing window and drag it around; you see a variety of solutions. How do they compare with what you drew earlier? (b) A separatrix is a curve such that above it solutions behave (as x increases) in one way, while below it solutions behave (as x increases) in quite a different way. There is a separatrix for this equation such that solutions above it grow without bound (as x increases)while solutions below it eventually decrease (as x increases). Use the applet to find its graph, and submit a sketch of your result. (c) Suppose y(x) is a solution to this differential equation whose graph is tangent to the m = −1 isocline: it touches the m = −1 isocline at a point (a, b), and the two curves have the same slope at that point. Find this point on the applet, and then calculuate the values of a and b. (d) Now suppose that y(x) is a solution to the equation for which y(a) < b, where (a, b) is the point you found in (c). What happens to it as x → ∞? More specifically, give an explicit function f(x) whose graph is asymptotic to the graph of the function y(x). For large x, is y(x) > f(x), y(x) < f(x), or does the answer depend on the value of y(a)? The
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