Elementary Linear Algebra and Dierential Equations
MAT 224

Spring 2014
MAT 224  SPRING 2014
Homework 2
Ryan Szypowski
Due April 17, 2014
1. Show that the following equations are not exact, then nd an integrating factor that
make them exact, then nd the general solution:
(a) (y + 2x) sin y) + (2(xy + x2 ) cos y + x sin y)y =
Elementary Linear Algebra and Dierential Equations
MAT 224

Winter 2013
MAT 224  WINTER 2013
Homework 7 Solutions
Ryan Szypowski
Due Never
Find the solution to the following systems of dierential equations:
1.
x
y
= 5x + 3y
= 9x + 7y
Solution: The matrix
5 3
9 7
A=
has eigenvalues of = 2, 4 with eigenvectors
1
1
v=
,
1
3
,
r
Elementary Linear Algebra and Dierential Equations
MAT 224

Winter 2013
MAT 224  WINTER 2012
Homework 4
Ryan Szypowski
Due Feb 21, 2013
1. Find an annihilator for each of the following functions.
(a) 4 + 7x e3x
(b) 8xe4x cos(2x) e4x sin(2x)
(c) 2e2x + 2e2x cos(2x)
2. Find the general solution to the following dierential equa
Elementary Linear Algebra and Dierential Equations
MAT 224

Winter 2013
MAT 224  FALL 2012
Ryan Szypowski
Method of Undetermined Coecients with Annihilators
In order to solve P (D)y = f (x) where P (D) is a linear dierential operator with constant
coecients (and hence P is a polynomial) we can perform the following steps:
1.
Elementary Linear Algebra and Dierential Equations
MAT 224

Winter 2013
MAT 224  FALL 2012
Worksheet 2
Ryan Szypowski
1. Answer the following word problems.
(a) A tank is originally lled with 1L of solution with a concentration of 1g/L salt. If
a new solution with concentration 2g/L is poured in at an increasing rate given
b
Elementary Linear Algebra and Dierential Equations
MAT 224

Winter 2013
MAT 224 WINTER 2013
Name:
Test #1 Solutions
Please show all of your work. Answers without justifaction may be worth 0.
Please make your answers easy to read. This means it should be clear what you are doing
from one step to the next and your work should b
Elementary Linear Algebra and Dierential Equations
MAT 224

Winter 2013
IE429Simulation
Course ProjectSpring 2016
3. At the Pilot Pen Company, a molding machine produces pen barrels of two different colorsred
and bluein the ratio of 2:3. The molding time is triangular (4, 5, 7) minutes per barrel. The barrels
go to a fillin
Elementary Linear Algebra and Dierential Equations
MAT 224

Winter 2013
California State Polytechnic University, Pomona
Degree Curriculum Sheet
Plan (Major) INDUSTRIAL ENGINEERING
Subplan/Option
20132014
Catalog Year
Minimum Units Required
198
Required Core Courses
Course
Units
Fundamentals of Human Factors Engineering
Eleme
Elementary Linear Algebra and Dierential Equations
MAT 224

Winter 2013
MAT 224  WINTER 2013
Homework 6
Ryan Szypowski
Due March 7, 2013
1. Consider the following linear system of equations:
2x1 + 3x2 + 4x3 = 3
4x1 + 2x2 + x3 = 3
2x1 + x2 + 3x3 = 4.
(a) Write down the augmented matrix for this system.
(b) Reduce this matrix
Elementary Linear Algebra and Dierential Equations
MAT 224

Winter 2013
MAT 224  WINTER 2012
Homework 5
Ryan Szypowski
Due Feb 28, 2013
1. Find the general solution to the following dierential equation
y + y = sec x.
2. First prove that
y1 = 1, y2 = x2
is a fundamental set of solution for
x2 y + 3xy = 0.
Then, use this to nd
Elementary Linear Algebra and Dierential Equations
MAT 224

Spring 2014
MAT 224  SPRING 2014
Homework 4
Ryan Szypowski
Due May 8, 2014
1. Let
2 0 1
A = 1 1 0 ,
0 2 1
1 2
2 ,
B= 4
1 1
3
x = 1 ,
2
2 0 1
C = 1 1 0 .
0 2 0
(a) Compute det(A). Is A singular or nonsingular?
(b) Compute Ax.
(c) Compute AB and BA if they make sense.
Elementary Linear Algebra and Dierential Equations
MAT 224

Spring 2014
MAT 224  SPRING 2014
Homework 2 Solutions
Ryan Szypowski
Due April 17, 2014
1. Show that the following equations are not exact, then nd an integrating factor that
make them exact, then nd the general solution:
(a) (y + 2x) sin y) + (2(xy + x2 ) cos y + x
Elementary Linear Algebra and Dierential Equations
MAT 224

Spring 2014
MAT 224  SPRING 2014
Homework 1
Ryan Szypowski
Due April 10, 2014
1. Determine the general solution to the following dierential equations:
(a) xy dx + (x + 1) dy = 0
(b)
dy
dx
=
(c) y +
y 2 +2xy
x2
1
y = ex
x
2. Determine the solution to the following in
Elementary Linear Algebra and Dierential Equations
MAT 224

Spring 2014
MAT 224  SPRING 2014
Homework 3
Ryan Szypowski
Due April 24, 2014
1. For each of the following autonomous dierential equations do each of the following:
(i ) nd and classify all equilibria, (ii ) provide a phaseline diagram including all of
the usual an
Elementary Linear Algebra and Dierential Equations
MAT 224

Spring 2014
MAT 224  SPRING 2014
Homework 1 Partial Solutions
Ryan Szypowski
Due April 10, 2014
1. Determine the general solution to the following dierential equations:
(a) xy dx + (x + 1) dy = 0
Solution: We begin by manipulating the equation as
dy
x dx
=
.
y
x+1
I
Elementary Linear Algebra and Dierential Equations
MAT 224

Spring 2014
MAT 224  SPRING 2014
Homework 5
Ryan Szypowski
Due May 15, 2014
1. For each of the following sets, determine if the functions are linearly dependent or
independent.
(a) cfw_1, ex , sin x
(b) cfw_1 x, 1 + x
(c) cfw_e2x , e2x cos(3x), e2x sin(3x)
2. For ea
Elementary Linear Algebra and Dierential Equations
MAT 224

Winter 2013
MAT 224  WINTER 2012
Homework 3 Additional Problems
Ryan Szypowski
Due Feb 5, 2013
1. Sketch the phase line diagram for the following autonomous ODEs and then discuss
the behaviour as t for the given initial condition.
(a) y = (y 1)(y + 1),
(b) y = ey 1,
Elementary Linear Algebra and Dierential Equations
MAT 224

Winter 2013
MAT 224  FALL 2012
Worksheet 1
Ryan Szypowski
1. Show that the functions given below are solutions to the given dierential equations
or initial value problems.
1
, y + 2yy = 0, y(1) = 1
x
(b) y = x2 + Cx + D, y = 2
(a) y1 =
2. Determine the solutions to
Elementary Linear Algebra and Dierential Equations
MAT 224

Winter 2013
CALIFORNIA STATE POLYTECHNIC UNIVERSITY, POMONA
INDUSTRIAL AND MANUFACTURING ENGINEERING
Production Planning and Control
IME 3261
Dr. K. Abedini
[email protected]
Office 172639
Phone: (909) 862569
Fax: (909) 8692564
SCHEDULE
Date
621
6/23
6/28
6