CHAPTER CHAPTER
9
Discrete Dynamical Systems
9.1
Iterative Equations First-Order Iterative Equations
an - 1 For Problems 1-12 we use the fact that the solution of yn +1 = ayn + b is yn = a n y0 + b . a -1
1.
yn +1 = yn + 3
1 2
1 2
y0 =
MA232 Fall 2008 Test 2 Practice October 3, 2008 The answers to the practice problems below will be posted on the website soon.
1
Variation of Parameters (Euler-Lagrange Method)
Solve the following problems from pg.71 using Variation of Parameters. Use the
MA232 Fall 2008 Test 2 Practice Solutions October 7, 2008
1
Variation of Parameters (Euler-Lagrange Method)
pg.71 problem 22: y=e-2t pg.71 problem 23: y=-2e-2t + et pg.71 problem 24: y= e +e 2
t 3t
pg.71 problem 26: y=e-t ln |1 + et | - ln |2|e-t pg.71
Conservation
Ordinary Dierential Equations
Lec 7, Sec 2.4, Mixing Problems and Newtons Law of Cooling
Guangming Yao
Clarkson University
Jan 24, 2014
Conservation
Mixing Problems
Conservation
Mixing Problems
Work with amounts (not concentrations)
Principle
MA232 Fall 2008 Test 3 Practice November 2, 2008 The answers to the practice problems below will be posted on the website soon.
1
Method of Undetermined Coefficients
Solve problems 22, 24, 26-28, 34, 36, 43, 45, and 49 from pgs. 253-254. I would start wit
MA232 Fall 2008 Test 1 Practice Solutions September 5, 2008
1
Classification
pg.62 #2: dependent variable is y, independent variable is t, 1st order. ODE, linear, variable coefficient equation, non-homogeneous pg.62 #4: dependent variable is y, independen
MA232 Fall 2008 Test 1 Practice September 5, 2008 The answers to the practice problems below will be posted on the website soon.
1
Classification
What is the order of the differential equation? What is/are the dependent variable(s)? What is/are the indep
MA232 Numerical Project
Fall 2008
Due 9/19
1
Reduction of Order
The numerical methods that we will look at this week are good for solving 1st order DE's. However, we wish to model 2nd order DE's because they are more interesting. This is O.K. because we c
MA232 Fall 2008 Final Study Guide December 1, 2008 Much of the practice below has been taken from old practice problems, class problems, and homework. There is no practice Final, but I encourage you to create your own from the following problems. To do th
MA232 Fall 2008 Final Study Guide Solutions December 1, 2008 Here are the solutions to even problems and others not in the book. Also, the systems solutions are written here in the format in which the test problems should be answered.
1
Variation of Param
First Order Linear Equations
Ordinary Dierential Equations
Lec 6, Sec 2.2, Integrating Factors
Guangming Yao
Clarkson University
Jan 22, 2014
First Order Linear Equations
Clicker Quiz
Which dierential equation is a linear equation?
C:
y y 2 = 0.
y + 3 y 2
Linear Equations
Ordinary Dierential Equations
Lec 5, Sec 2.1, Linear Equations
Guangming Yao
Clarkson University
Jan 20, 2014
Linear Equations
Clicker Quiz
Determine the solution to the dierential equation
y
A:
B:
C:
D:
y
y
y
y
2
1
= 2 ke t .
= ke t .
2
Growth and Decay
Ordinary Dierential Equations
Lec 4, Sec 2.3, Growth and Decay
Guangming Yao
Clarkson University
Friday, Jan 17, 2014
Growth and Decay
Growth and Decay
Rate of change(growth/decay) is proportional to the
amount that is present.
amount pre
Solutions to DEs and Slope Field
Ordinary Dierential Equations
Solutions to DEs and Slope Field
Guangming Yao
Clarkson University
Monday, 13 Jan 2014
Solutions to DEs and Slope Field
Outline
1
Solutions to DEs and Slope Field
Solutions to DEs
Slope Fields
Ma 232
Review of Freshman Calculus Basics
Spring 2014
Differentiation formulas:
f ( x) and g ( x) differentiable, c a constant:
d
d
cf ( x) cf c( x).
f ( x) r g ( x) f c( x) r g c( x).
dx
dx
d
d f ( x) f c( x) g ( x) f ( x) g c( x)
.
f ( x) g ( x) f c(
Separable Equations
Ordinary Dierential Equations
Lec 3, Sec 1.3 Separabel Equations
Guangming Yao
Clarkson University
Wed, 15 Jan 2014
Separable Equations
Separable Equations
How do we nd a solution to a DE?
Method 1: Separation of Variables
Works for Se
Introduction
Ordinary Dierential Equations
Introduction
Guangming Yao
Clarkson University
Friday, 10 Jan 2014
Introduction
Outline
1
Introduction
Syllabus
What is a Dierential Equation
Modeling
Introduction
Syllabus
Read the syllabus(on moodle)! The sylla
MA232 Dierential Equations, Spring 2014
Topics: First Order Equations:
Linear/Nonlinear Equations, Laplace Transforms
Second Order Equations: Linear Equations, Cauchy-Euler Equations, Laplace Transforms
Systems of Equations:
Linear Algebra, Eigen Systems