Math 229
More notes on linear systems
April 19, 2009
We are discussing homogeneous linear systems. That is, we are discussing
systems of the form
~ 0 (t) = A~ (t)
x
x
(1)
where A is an n n matrix and ~ represents a function from R to Rn : The
x
fundamenta
Math 229
Solutions to HW #8
April 14, 2009
Section 7.5, #15, 17, 31
#15 We are given the equation (or system of equations, if you prefer):
5
3
~0 =
x
1
1
~
x
We calculate the characteristic polynomial of this matrix:
p (x)
=
5
det
= (5
= x2
x
3
1
x
1
x) (
Math 229
February 17, 2009
Solutions to HW #4
Section 2.5, #3, 9, 16
Section 2.6, #2, 4, 5, 7
Section 2.5, #3, 9, 16
3. We are given the autonomous equation
y 0 = y (y
1) (y
2)
If we let f (y) denote the right hand side:
y 0 = f (y)
we can plot f against
Math 229
Feb10, 2009
Solutions to HW #3
Section 2.3, #3, 8, 10, 16
Section 2.8, #6a,c, 8a, 10a
Section 2.3, #3, 8, 10, 16
3. We are to perform two operations in succession, and determine the net
eect. We have a tank of 100 gallons of pure water, with wate
Math 229
Solutions to HW#6
March 3, 2009
Section 3.3, #15, 17, 19, 24, 27
#15 We are given the equation
t2 y 00
t (t + 2) y 0 + (t + 2) y = 0
which can be written in standard form
y 00
t+2 0 t+2
y + 2 y = 0;
t
t
where we restrict to the interval y > 0: Ac
Math 229
February 3, 2009
Solutions to HW #2
Section 1.2, problems 4, 5,
Section 2.1, #13, 15, 30,
Section 2.2, #13, 21, 23
Section 1.2, problems 4, 5
#4. a. The equation
y 0 = ay
b
where a and b are constants and a 6= 0, can be solved as we did in class.
Math 229
Solutions to HW #9
April 21, 2009
Section 7.6, #2, 6
#2. We are given the system
1
1
~0 =
x
4
1
~
x
We calculate the characteristic polynomial of this matrix:
1
det
x
4
1
1
=
x
2
( 1
x) + 4
= x2 + 2x + 5
so we get eigenvalues
p
2
16
2
1
( 1 + 2i)
Math 229
Feb 27, 2009
Notes on linear algebra
Some basic ideas from linear algebra play an essential role in our subject,
so I oering this summary as a reference. These notes won substitute for
m
t
a proper textbook. In particular, I won give proofs of an
Math 229
Notes on linear systems
April 10, 2009
We have been discussing the behavior of solutions to the linear system
~ 0 (t) = A~ (t)
x
x
(1)
where A is a constant n n matrix. The presentation I gave in out latest class
is not quite the same as that of
Math 229
Feb 18, 2009
Notes on autonomous equations
In class on February 12th we discussed autonomous equations, Some of the
issues we discussed do not appear in our text (I think) and were discussed only
very brie by us, so I am writing these notes to gi
Math 229
Solutions to HW #10
April 28, 2009
Sec 7.7, #3, 5, 11, Sec 7.8, #1, 2, 7
Sec 7.7, #3, 5, 11
#3. We are given the system
2
3
~0 =
x
1
2
~
x
To nd a fundamental matrix we must nd two independent solutions, and use
them as the columns of the fundame
Math 229
January 29, 2009
Solutions to HW #1
Section 1.1
Problems 15-21
We are given 10 equations and six pictures of direction elds, and we are
supposed to nd, for each picture, the equation that generates it. Each equation
has the form y 0 = f (y) ; whe