Document Sample

Math 350 September 9, 2010 Homework Assignment #2 due Thursday, September 16, 2010 1. (i) Find the critical points and phase portrait of the autonomous first-order DEs given below. (ii) Classify each critical point as stable, unstable or semi-stable. (iii) Then by hand, sketch a typical solution curve in the regions in the ty-plane determined by the graphs of the equilibrium solutions. dy a. y2 3y dt dy y 2 4 b. dt y2 4 y2 dy c. dt 2. Consider the autonomous DE dy / dt f ( y) where the graph of f is given below. (i) Use the graph to locate the critical points of each DE. (ii) Sketch a phase portrait of the DE. (iii) By hand, sketch a typical solution curve in the sub-region of the ty-plane determined by the graphs of the equilibrium solutions. a. b. 1 Math 350 September 9, 2010 dy 3. Suppose we know that the graph below is the graph of a solution to f (t ). dt a. How much of the slope field can you sketch from this information? b. What can you say about the solution with y(0) 2? [For example, can you sketch the graph of this solution? dP 4. The following DE is a well-known population model kP h where h and k are positive dt constants. For what initial values P (0) P0 does this model predict that the population will go extinct? 5. The autonomous differential equation is a model for the velocity v of a body of mass m that is falling under the influence of gravity, modified so that air resistance is proportional to v2. dv m mg kv 2 dt Use a phase portrait to find the terminal velocity of the body, and explain your reasoning. 6. a. Consider the autonomous DE dy dt y 2 y 6 . Find intervals on the y-axis for which solution curves are concave up and intervals for which solution curves are concave down. b. Discuss why each solution curve of an initial-value problem of the form dy dt y 2 y 6 where 2 y0 3, has a point of inflection with the same y-coordinate. What is that y-coordinate? c. Carefully sketch the solution curve for which y (0) 1. d. Carefully sketch the solution curve for which y (2) 2. 7. When certain kinds of chemicals are combined, the rate at which the new compound is formed is dX modeled by the autonomous differential equation k ( X )( X ), where k > 0 is a constant of dt proportionality and 0. Here X (t ) denotes the number of grams of the new compound formed in time t. a. Use the phase portrait of the DE to predict the behavior of X (t ) ast . 2 Math 350 September 9, 2010 b. Consider the case when . Use a phase portrait of the DE to predict the behavior of X (t ) ast when X (0) . Then when X (0) . c. Verify that an explicit solution of the DE in the case when k = 1 and is 1 X (t ) . (t c) d. Find a solution that satisfies X (0) / 2. e. Find a solution that satisfies X (0) 2 . f. Graph the solutions you found in (d) and (e). Does the behavior of your solutions as t agree with your answer to (b)? dP 8. Suppose you wish to model a population with a differential equation of the form f ( P), where dt P(t ) is the population at time t. Experiments have been performed on the population that give the following information: The population P 0 remains constant A population close to 0 will decrease A population of P 20 will increase A population of P 0 will decrease a. Sketch the simplest possible phase line that agrees with the experimental information given.] b. Give a rough sketch of the function f ( P ) for the phase line in part (a). c. What other phase lines are possible? 9. The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time t. After 3 hours it is observed that 400 bacteria are present. After 10 hours 2000 bacteria are present. What was the initial number of bacteria? 10. According to Newton’s empirical law of cooling/warming, the rate at which the temperature of a body changes is proportional to the difference between the temperature of the body and the temperature of the surrounding medium (the ambient temperature). If T(t) represents the dT temperature of a body at time t, Tm the temperature of the surrounding medium and the rate at dt which the temperature of the body changes, then Newton’s law of cooling/warming translate into the dT mathematical statement k (T Tm ), where k is a constant of proportionality that is less than zero. dt Suppose a dead body was found within a closed room of a house where the temperature was a constant 70F. At the time of discovery the core temperature of the body was determined to be 85F. On hour later a second measurement showed that the core temperature of the body was 80F. Assume that the 3 Math 350 September 9, 2010 time of death corresponds to t = 0 and that the core temperature at that time of death was 98.6F. Determine how many hours elapsed before the body was found. 11. A mathematical model for the rate at which a drug disseminates into the bloodstream is given by dy r ky, where r and k are positive constants. The function y (t ) describes the concentration of the dt drug in the bloodstream at time t. a. Use a phase portrait to find the limiting value of y(t ) as t . b. Solve the DE subject to y (0) 0. Sketch the graph of y (t ) and verify your prediction in (a). c. At what time is the concentration ½ of the limiting value? 12. Eight differential equations and four phase lines are given below. Determine the equation that corresponds to each phase line and state briefly how you know your choice is correct. dt dt dt dt (i) y2 y 1 (ii) y y 1 (iii) y y3 (iv) y3 y dt dt dt dt dt dt dt dt (v) 2 y y2 (vi) y2 2 y (vii) y2 y (viii) y y2 dt dt dt dt (a) (b) (c) (d) 13. Perform Euler’s Method, with the given step size t on the given initial-value problem over the time interval specified. Your answer should include a table of approximate values of the dependent variable. You should also include a graph of your solution. (I suggest using Excel to plot the solution). dt a. t y 2 , y (0) 1 0 t 1 t 0.25 dt d b. 3 1 , (0) 1 0 t 5 t 0.5 dt dvc V (t ) vc 14. Reconsider the RC circuit equation . Suppose V (t ) 2cos(3t ) (the voltage source dt RC V(t) is oscillating periodically). If R = 4 and C = 0.5, use Euler’s method to compute the solution over the time interval 0 t 10 for the initial condition vc (0) 1. 4 Math 350 September 9, 2010 5

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 12 |

posted: | 3/8/2012 |

language: | English |

pages: | 5 |

OTHER DOCS BY xiuliliaofz

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.