MATH 308
Notes
Chapter 2, First Order Equations
Section 2.5, Autonomous equations, population dynamics
First order autonomous equations
y = f (y)
But in this chapter, it is mainly the example of the logistic equation which will
be studied.
The concepts to

Math 311: Topics in Applied Math 1
6: Eigenvalues
6.1: Eigenvalues and Eigenvectors
Solution
Determine eigenvalues.
det (A I) = 0
6
4
= 0
3
1
Summary
Let A be an n n matrix. If Ax = x for a scalar
and a nonzero vector x, then is an eigenvalue
associat

Math 311: Topics in Applied Math 1
5: Orthogonality
5.5: Orthonormal Sets
Let S be a subspace of V with orthonormal basis
cfw_u1 , . . . , un and let x V . If p = ci ui where
ci = x, ui , then p x S . Moreover, p is the
element of S that is closest to x

Math 311: Topics in Applied Math 1
5: Orthogonality
5.4: Inner Product Spaces
THEOREM: If V is an inner product space, then
v =
v, v for all v V denes a norm on V .
For vectors x and y in a normed linear space, the
distance between x and y is x y .
Summ

Math 311: Topics in Applied Math 1
5: Orthogonality
5.1: The Scalar Product in Rn
Summary
Let x, y
Solution
Since x and y be linearly independent, neither vector is a
multiple of the other. Hence the angle between them is
not a multiple of . Hence |cos |

Math 311: Topics in Applied Math 1
5: Orthogonality
5.3: Least Squares Problems
Solution
The coefcient matrix and right-hand side vector are
1
1 1
1
1 1
A=
0 1 1 ,
1
0 1
Summary
Recall an overdetermined system: Ax = b where A
is an m n matrix with m

Math 311: Topics in Applied Math 1
4: Linear Transformations
4.3: Similarity
202/1c
Let E = cfw_e1 , e2 be the standard basis for V = R2 and
F = cfw_u1 , u2 be another basis for R2 where
Summary
u1 =
Let E = cfw_v1 , . . . vn and F = cfw_w1 , . . . ,

Math 311: Topics in Applied Math 1
5: Orthogonality
5.2: Orthogonal Subspaces
Examples
233/1c
4 2
1
3
, determine a basis for
Summary
For the matrix A =
2
1
3
4
Subspaces X and Y of Rn are orthogonal if xT y = 0
each of the subspaces R AT , N (A), R

Math 311: Topics in Applied Math 1
3: Vector Spaces
3.6: Row Space and Column Space
166/11
Let A be an m n matrix. Prove rank (A) min (m, n).
Solution
Summary
The rank of A is the dimension of the row space of
A. Thus rank (A) m, the number of rows of A.

Math 311: Topics in Applied Math 1
5: Orthogonality
5.6: The Gram-Schmidt
Orthogonalization Process
Solution
2
5
and v2 =
.
1
10
They are linearly independent and hence form a
basis for the column space of A, although not an
orthonormal one.
The norm of

Math 311: Topics in Applied Math 1
6: Eigenvalues
6.3: Diagonalization
Examples
336/1b
5
6
2 2
XDX1 where D is diagonal.
Factor the matrix A =
Summary
into a product
k
THEOREM: If j j=1 are distinct eigenvalues of
an n n matrix A, their corresponding eig

MATH 308 - 300QT
Notes
Chapter 1, Introduction
Section 1.1 Mathematical models, direction fields
A dierential equation is an equation with a derivative in it.
Modeling: Finding a dierential equation whose solution is like some physical
phenomenon.
Some di

MATH 308
Notes
Chapter 3, Second Order Equations
Section 3.1, Homogeneous Equations with Constant Coecients
The general linear second order dierential equation is
P (t)y + Q(t)y + R(t)y = G(t)
Definition
The above equation is called homogeneous if G(t) =

MATH 308
Notes
Chapter 2, First Order Equations
Section 2.4, Dierences between linear and nonlinear equations
Existence and uniqeness
Theorem for the existence and uniqueness of the solution of linear rst order
ODEs
If the functions p and g are continuous

MATH 308
Notes
Chapter 2, First Order Equations
Section 2.6, Exact equations
Definition
The dierential equation
M (x, y) + N (x, y)y = 0
is exact if there is a function (x, y) for which
M (x, y) =
(x, y)
x
N (x, y) =
Example
2x + y 2 + 2xyy = 0
is exact:

MATH 308
Notes
Chapter 2, First Order Equations
Section 2.3, Modeling with first order equations
Situations with rates of change:
1. change of the concentration of some substance dissolved in a liquid
2. change of amount of money due to interest
3. change

MATH 308
Notes
Chapter 2, First Order Equations
Section 2.2, Separable equations
First order nonlinear equations
dy
= f (x, y)
dx
(x)
Special case: f (x, y) = M
N (y)
dy
or M (x) + N (y) dx
=0
Separation of variables
The solution of the ode
f (x, y) =
M

MATH 308 - 300QT
Notes
Chapter 1, Introduction
Section 1.2 Solutions of some Dierential Equations
Solutions of simple equations
What is a solution of a dierential equation?
A function if substituted in the left and right of the equation yields an equality

MATH 308
Notes
Chapter 2, First Order Equations
Section 2.1, Linear equations: Integrating factors
First order linear equations in standard form
dy
+ p(t)y = g(t)
dt
Solution
Let (t) = e p(t)dt (the integrating factor)
Integrating factor
Let p and g be c

Math 311: Topics in Applied Math 1
4: Linear Transformations
4.1: Denition and Examples
1. L : R2 R2 dened by L (x) = 3x is a linear
operator. A given vector is stretched by a factor of 3.
[More generally, for > 0, L (x) = x stretches
or shrinks a vector

Math 311: Topics in Applied Math 1
4: Linear Transformations
4.2: Matrix Representations of
Linear Transformations
Solution
Let A =
Ax =
Summary
1
1 0
. Then for each x R3 ,
0 1 1
x1
1
1 0
x2 x1
x2 =
= L (x) .
0 1 1
x3 x2
x3
If L : Rn Rm is a linear tra

Spring 2004 Math 253/501503
14 Vector Calculus
14.6 Parametric Surfaces & Areas
c 2004, Art Belmonte
Thu, 08/Apr
Solution
Well, x = 4 y 2 2z 2 for (y, z) D = (y, z) : y 2 + 2z 2 4
is one representation. Its graph, however, looks pinched. Were
trying to t

Spring 2004 Math 253/501503
14 Vector Calculus
14.7 Surface Integrals
c 2004, Art Belmonte
Tue, 13/Apr
Table Notes
1. Other rst-order moments are symmetrically dened.
Mxz =
Mxy =
y dS,
z dS
S
S
2. Other second-order moments are symmetrically dened.
Summar

Spring 2004 Math 253/501503
14 Vector Calculus
14.5 Curl and Divergence
c 2004, Art Belmonte
Thu, 08/Apr
Solution
The curl of F is curl F =
i
x
x2y
Summary
k
y
yz 2
z
x 2z
= 2yz, 2x z, x 2 .
F or
, ,
x 2 y, yz 2 , x 2 z = 2x y + z 2 + x 2 .
x y z
90

Math 311: Topics in Applied Math 1
1: Matrices & Systems of Equations
1.1: Systems of Linear Equations
Examples
These problems are exercises from the section in your
textbook, referenced by page and exercise number.
Summary
System of linear equations: A

Spring 2004 Math 253/501503
14 Vector Calculus
14.3 The Fundamental Theorem for
Line Integrals
c 2004, Art Belmonte
Tue, 06/Apr
2. The line integral C F dg is independent of path if and only
if C F dg = 0 for every closed path in E.
3. Let F be continuous

Math 311: Topics in Applied Math 1
1: Matrices & Systems of Equations
1.3: Matrix Arithmetic
Zero matrix: An m n matrix all of whose entries
are zero; denoted O (uppercase bold letter O) or 0
(bold number zero). It is the additive identity of the
set of

Spring 2004 Math 253/501503
14 Vector Calculus
14.2 Line Integrals
c 2004, Art Belmonte
Thu, 25/Mar
Table Notes
1. Other rst-order moments are symmetrically dened.
Mxz =
y ds,
Mxy =
C
z ds
S
2. Other second-order moments are symmetrically dened.
Summary
x