Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
Higher Order Linear
Differential Equations
Chap 4.4
1
Recap & Comments
3
Superposition
4
Superposition (cont)
5
6
Superposition Principal for Linear
DEs
Suppose
is a solution to
Then
is a solution to
7
8
Standard Form
Where is of the form
Polynomials in
Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
Linear Algebra
Matrix Sum and Products
Chap 3.2
1
Recap & Comments
2
3
Elementary Row Operations
Interchange row i and row j
Multiply row i by a constant
Add c times row j to row i
(leaving row j unchanged)
5
6
7
Row Reduced Echelon Form
(RREF)
A matrix i
Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
Linear Algebra
Matrix Sum and Products
Chap 3.1, 3.2
1
Recap & Comments
2
3
4
Matrix
5
6
7
8
9
Matrix Addition and Scalar
Multiplication Properties
A, B, C are mxn matrices then:
A+B is an mxn martix
A+B = B+A
A+(B+C) = (A+B)+C
c(kA) = (ck)A
A+0 = A
A+(A
Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
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Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
February 6, 2014
MAT 485 Section 3, Quiz 3
NAME: Fk. A Corgi? Student ID: 6? 4.2 (7 (75;,
Instructions: Write your answers and show all your work: no work, no credit! No books, notes,
or electronic devices are allowed. There are 2 questions on 2 pag
Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
Higher Order Linear
Differential Equations
Method of Variation of
Parameters
Chap 4.5
1
Recap & Comments
4
5
Standard Form
Standard
Form
Realistically:
Solving Using
Undetermined Coefficients
1.
Find the homogeneous solution
a. (4.1) Two real roots
b. (4
Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
First Order DEs
Linear Equations
Chap 2.3, 2.4
1
Recap & Comments
2
Growth and Decay
For
each k, the solution to the IVP
is given by
Solution curves are growth curves
for and decay curves for
3
4
Example
Carbon dating uses the decay of C14. The halflif
Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
Complex Arithmetic
Background
Definitions
Complex number any number of the
form
Addition, Subtraction,
Multiplication
Addition/Subtraction
Multiplication
Complex Conjugate
Complex Conjugate of is defined to
be:
Modulus of a complex number, , is
written
Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
Linear Algebra
Matrix Sum and Products
Chap 3.2
1
Recap & Comments
2
3
Elementary Row Operations
Interchange row i and row j
Multiply row i by a constant
Add c times row j to row i
(leaving row j unchanged)
5
6
7
Row Reduced Echelon Form
(RREF)
A matrix i
Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
First Order DEs
Linear Equations
Chap 2.1, 2.2
1
Recap & Comments
2
Today
3
Linear Differential
Equations
A equation
is linear if it is of the form
Where all functions of t are assume to
be defined over some common interval
I.
If over the interval I, the
Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
Linear Algebra
Eigenvalues and
Eigenvectors
Chap 5.3
Recap & Comments
Eigenvalue and Eigenvector
Let
be a linear transformation from
vector space into vector space . A
scalar is an eigenvalue of if there is a
nonzero vector such that
Such a nonzero vector
Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
Linear Algebra
Matrix Inverse
Chap 3.5, 3.6
1
Recap & Comments
2
3
Vector Subspace Theorem
A nonempty subset of W of a vector space
then V is a subspace of V it is closed
under addition and scalar multiplication:
i. If then
ii. If then
7
Definitions
Span
Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
Linear Algebra
Inverses, Determinants and
Cramers Rule
Chap 3.3, 3.4
1
Recap & Comments
2
Matrix Inverse
If there
exists, for an nxn matrix ,
another matrix of the same order
such that
then is called the inverse of
matrix and is said to be
invertible.
3
4
Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
Linear Algebra
Inverses, Determinants and
Cramers Rule
Chap 3.3, 3.4
1
Recap & Comments
2
Matrix Inverse
If there
exists, for an nxn matrix ,
another matrix of the same order
such that
then is called the inverse of
matrix and is said to be
invertible.
3
4
Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
Higher Order Linear
Differential Equations
Chap 4.1, 4.2, 4.3
1
Recap & Comments
2
Spring & Mass
4
Simple Harmonic Oscillator
The
simple harmonic oscillator
equation is
a secondorder nonhomogeneous linear
differential equation with constant
coefficients
Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
Linear Algebra
Linear Transformations
Chap 5.1, 5.2
Recap & Comments
Linear Transformation
A linear
transformation T on a vector
space V to a vector space W is a
function T : V W that preserves scalar
multiplication and vector addition.
That is, for all V
Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
MAT 485
Paul Tuttle
1
Blackboard Structure
Administrative Contains all administrative
related material to include the Syllabus and
Calendar
Background Calculus summaries, useful tables,
etc.
Classes Handouts from class
Exams Preparation material, Answ
Differential Equations and Matrix Algebra for Engineers
MAT 485

Spring 2014
First Order DEs
Linear Equations
Chap 2.5
1
Recap & Comments
2
Analytic Definition of Solution
Analytically, y(t) is the solution of
differential equation (1) if substituting
y(t) for y reduces to the identity
y(t) f(t, y(t)
On an appropriate domain for t
Differential Equations and Matrix Algebra for Engineers
MAT 485

Fall 2010
FINAL EXAMINATION
MAT 485 Spring 2009
NAME:
(write in capitals)
NAME:
L.
(8 points.) Suppose a tank has 100 gal of water in which 20 lb of salt
has been dissolved. Brine of concentration2lblgal is pumped into the tank
at a rate of 5 gal/min. The brine is