Sample Exam
I M242 Spring 2008
SHOW ALL WORK!
- Find solutions to each
initial value problem.
a) 
, 
b) 
, 
- a) Sketch the slope
field for the equation
on the intervals
and 
. Fill out the sketch with numerous solution curves to
form a phase portrait.
b) What are the possible long-term values for y
(that is, what are the possible values of 
)? Which initial conditions 
go to which long-term
y values?
c) If 
, estimate the value of 
using Euler’s method
with stepsize 0.1. Show two steps by hand. Repeat using stepsize 0.01, and
report only the digits that you believe are accurate. What is the long-term
value for y for this initial condition? Graph this solution curve on
your phase portrait from a) (make it darker than the other solution curves).
- For the differential
equation
with 
:
a)
Find
the equilibrium points.
b)
Sketch
the phase line and classify the equilibrium points as sink, source or node.
c)
Now
drop the assumption that a must be greater than 1. Sketch phase lines
for 
and display them next
to each other in order (equally spaced). What is the bifurcation point (in
terms of a) and how does the number and type of equilibrium point change
at the bifurcation point?