Homework, MATH 414, Fall 2004
Homework is from our text unless indicated).
Homework is due in class on the day listed.
You
are welcome and encouraged to help each other in solving the
problems, but for homework which is turned in,
you must write your own solutions.
You will usually learn the most if you work on the homework alone
before consulting solutions or a classmate.
Homework #1 (Wed. Sept. 1)
p.8: 1 (solution)
p.9: 3 (solution)
Homework Assignment #2 (Wed. Sept. 8)
p.12: 5, 7
p.13: 11
p.18: 2, 3
Homework Assignment #3 (Wed. Sept. 15)
p. 25: 5abc (just for n=1)
p. 27: 7, 8
p. 29: 14
p. 30: 18
p. 31: 1,2
Homework Assignment #4 (Wed. Sept. 22)
p.39: 3, 4
[For #3: find
a number M such that M is greater than |phi(t)| for all t.]
p.41: 8
p.49:
23, 25, 27
[Hint for #25: let C be the matrix with first row (0,1) and
second row (1,0),
and consider Example 23.]
p.54: 5
Homework Assignment #5 (Friday Oct. 1)
p. 56: 4
[Hint for #4: for a given invertible nxn matrix T, the map f
defined by
f(M) = T^{-1}MT is a continuous
function on the space of nxn matrices.]
p. 58: 6
p. 59: 8,10
p. 70: 13
p. 71: 18, 20
[In #20: just do the example p=6, q=9]
Homework Assignment #6 (Monday Oct. 11)
p.73: 23, 26
p.78: 7
p.82: 3
p.88: 1-4
p.95: 18,19
Homework Assignment #7 (Monday Oct. 25)
Ch. 2 Misc. Ex: 1,12 (pg. 103 and 106)
Sec. 3.1: 1,9,10,14 (pg. 110,118,120)
Sec 3.2: 2, 3
[In #2: you do not have to give any proof, but give a
clear full statement.]
[In #3: The first of the three approximations to compute
is the constant function phi_0.]
Homework Assignment #8 (Monday Nov.1)
p.137: 1
Homework Assignment #9 (Monday Nov.8)
(Reformulate any higher order DE as a DE
of the form y'=f(t,y), and interpret
stability in terms of y. For example,
a DE involving variables x, x', x"
where x is scalar-valued
will become a DE involving y, where
y_1=x and y_2=x'.)
p. 150: 4
p.151: 9
p.154: 3-10
Homework Assignment #10 (Monday Nov.15)
Problem 1:
p.159, #17
Problem 2:
Let (x,y) denote the coordinates of a
vector in the plane.
Consider the
autonomous differential equation
(x,y)' = (1 - y , x^2 - y^2) .
-
(a) Determine the equilibrium solutions
(i.e. the points where (x,y)'=0).
-
(b) For each of the equilibrium points,
compute the 2x2 derivative matrix of f
at the point, and determine
if possible
the type of the equilibrium point
(saddle point, stable node, etc.) .
-
(c) Draw the phase portrait in the plane for
the differential equation.
(Take into
account the information from (b), and
the regions where x' and y' are positive and
negative.)
Homework Assignment #11 (Wednesday Nov. 24)
p. 214: Ex. 1
p. 215: Ex. 2,3
Homework Assignment #12 (Monday Dec. 6)
p. 219: Ex. 2. This is like Example 1. Do Ex.2 as follows:
-
Define V from (5.15).
-
Show V is positive definite on the open set of points y=(y_1,y_2)
for which y_1 > -3b/2.
-
Show that the origin is in the interior U of a region bounded by a
closed curve through the point (-b,0) on which V is constant.
- Show for that if phi is a solution and phi(t_0) is in U,
then phi(t) converges to the origin as t goes to infinity.
p. 227: Ex. 12
p. 250: Ex. 3ac