Math 21: Final. Friday, 06/03/2011

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1 Math 21: Final Friday, 06/03/2011 Complete the following problems. You may use any result from class you like, but if you cite a theorem be sure to verify the hypotheses are satisfied. When finished hand in exam to Eric or Jack. This is a closed-book exam. No other calculators or other electronic aids. You may bring in one 8.5 by 11 inch piece of paper filled with handwritten notes on both sides into the exam. Please staple these notes to your exam. If your notes do not fit this description please leave the exam room now and either recopy your note sheet or take the exam without your note sheet. In order to receive full credit, you must show all of your work and justify your answers. Your answer should be clearly labeled. There is scrap paper attached to the back of the exam. Please do not bring in outside pieces of paper. Please sign the following: On my honor, I have neither given nor received any aid on this examination. I have furthermore abided by all other aspects of the honor code with respect to this examination. Name: Signature: (1) (/15 points) (2) (/10 points) (3) (/10 points) (4) (/10 points) (5) (/10 points) (6) (/10 points) (7) (/10 points) Total. (/75 points) 1

2 (1) (15 points) Consider the following differential equation dy dx = y2 3y + 2. (a) (1 point) Classify this equation (i.e. what kind of DE is it?). (b) (2 points) Find and sketch all constant solutions in the xy-plane. (c) (2 points) Sketch the solution passing through y(0) = 1.5. For this solution, what is lim x y(x)? 2

3 (d) (10 points) Find all solutions of the differential equation dy/dx = y 2 3y + 2, showing all steps. 3

4 (2) (10 points) Solve the initial value problem for y = y(x): showing all steps. y = ln(x), y(1) = 2, xy 4

5 (3) (10 points) Find all functions y = y(x) that have the property that the slope of the tangent line at (x, y) is proportional to the product of the coordinates. 5

6 (4) (10 points) (a) (5 points) Determine conditions on a constant r so that y = cos(rt) is a solution to the differential equation 4 d2 y dt 2 = 25y. 6

7 (b) (5 points) For the values of r you found in (a), show that y = A cos(rt) + B sin(rt) is also a solution to the differential equation, where A and B are constants. 7

8 (5) (10 points) Consider a population P = P (t) with constant relative birth and death rates α and β respectively, and a constant emigration rate m, where α, β, m are positive constants. Assume α > β. The rate of change of population over time is modeled by dp = kp m, where k = α β. dt (a) (8 points) Find the solution of this equation satisfying P (0) = P 0, showing all steps. 8

9 (b) (2 points) What value of m leads to an exponential expansion of population? What value leads to a constant population? 9

10 (6) (10 points) Consider an alternate love affair between Romeo and Juliet. As in class, R(t) denotes a measure of Romeo s love for Juliet and J(t) measures Juliet s love for Romeo, (positive values indicate love, negative values indicate hate). Suppose the following system of differential equations models their love dj dt = J R dr dt = J + R (a) (2 points) Find the equilibrium points for this system. (b) (2 points) Assume R = R(J). Write an equation for dr dj. (c) (2 points) Using the techniques given in class, could you solve your equation for dr/dj? Why or why not? 10

11 (d) (2 points) Consider the phase portrait for the system given below (here Juliet is along the x axis). For the phase trajectory plotted, which direction do we travel along as t? Justify your answer with a computation and show your answer with an arrow. (e) (2 points) end? Given the model above, how does this version of Romeo and Juliet 11

12 (7) (10 points) True/False, short-answer. No justification needed. (a) True/False: If dp dt = 0.4P 0.01P 2, then the carrying capacity is M = 0.01/0.4. (b) True/False: y = x ln(x) is a solution to: x 2 y xy + y = 0. (c) True/False: dφ dθ = sin2 (θ) is a first order autonomous differential equation. (d) Short answer: Solve the differential equation y = 0. (e) Short answer: Given the DE, y = y y 2, find the equation of the tangent line to the solution passing through y(2) = 1. 12

13 (f) Consider the following direction field plots. Match each of the following differential equations to one of these fields or write NO MATCH if there is no match: 1. y = y 2 x 2. Match: 2. y = sin(2πx) sin(2πy). Match: 3. y = x 2. Match: 4. y = 15y(1 y) 2. Match: 5. y = 15y(1 y) Match: (a) Figure A (b) Figure B (c) Figure C (d) Figure D 13

14 (e) Figure E (f) Figure F Scrap paper 14

15 Scrap paper 15

Math 19 Practice Exam 2B, Winter 2011

Math 19 Practice Exam 2B, Winter 2011 Name: SUID#: Complete the following problems. In order to receive full credit, please show all of your work and justify your answers. You do not need to simplify your