# Homework2

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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.

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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  .

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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 

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

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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.

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Math 350   September 9, 2010

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