# doc ```MATH230
Differential Equations
Spring 2008
Homework assignment 8
Due Thursday, 03/27/08
(1) For each part, if possible, create an autonomous differential equation that has the
stated properties. Explain your thought processes as to how you created each
differential equation and include any and all graphs or representations that helped you
think about each task. If it is not possible to come up with a differential equation with
the stated properties provide a justification for why it cannot be done.
(a) Exactly three constant solution functions, two repellors and one attractor.
(b) Exactly two constant solution functions, one attractor and one node.
(c) Exactly two constant solution functions, both attractors.
dC
 0.4C  28 , how long will it take for
dt
a cup of hot coffee that is initially 180 F to cool down to 100 F? Use reverse
product rule to figure this out, and check the reasonableness of your answer with
Euler’s method.
(2) According to the rate of change equation
dC
 0.4C  28
dt
assumes that the room temperature remains constant. What if instead the room
temperature increased as the coffee cooled? Could the rate of change equation
dC
 0.4C  28 be modified to reflect this new assumption? Would we need a
dt
second differential equation for the room temperature? Discuss your ideas.
(3) In the previous coffee cooling problem, the differential equation

(4) For each part below you are provided with a graph of a rate of change equation rather
than the equation itself (Note that dy/dt depends only on y).
(a)
(b)
dy/dt
dy/dt
y
y
(i) Find and classify all equilibrium solutions as either an attractor, repellor, or node and
create a flow line for each that summarizes your results.
(ii) Describe the long-term behavior of every possible solution function / initial condition.
State your conclusions about the long-term behavior of different solutions in a few
sentences. Explain how you reached your conclusions in words and/or with suitable
graphs. Be brief, concise, and clear
```