AM351: ODE II
Spring 2008
2
AM 351 ODE II
Contents
1 Theory of higher order linear ODEs 1.1 Existence and uniqueness of solutions of nth order 1.2 Homogeneous ODEs and superposition theorem . 1.2.1 Linear operator . . . . . . . . . . . . . . . 1.
AM351 Spring 08
Due Wednesday, July 30th 2008
Assignment 4
Instructions: write your solutions clearly, explain your steps carefully and refer to theorems and lemmas when needed. The assignment is due either in class or in the assignments drop box (
Selected Topics in Applied
Mathematics
Charles L. Byrne
Department of Mathematical Sciences
University of Massachusetts Lowell
Lowell, MA 01854
September 13, 2012
(Supplementary readings for 92.530531
Applied Mathematics I and II)
(The most recent version
AMATH 351: Ordinary Dierential Equations II
or
Methods in Applied Mathematics I
F. J. Poulin
Contents
Contents
i
Preface
vii
1 Theory of Second-Order Linear DEs
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Dierent Forms of the
AM 351: Assignment 2:
Due 11:30am, Oct 5th, 2012
1. Heated spring Recall that the equation
my (t) + by (t) + ky (t) = 0
describes the position y of a mass attached to a wall by both a spring, with spring constant
k , and a damper, with damping constant b.
AM 351: Assignment 3
Due October 24th, 2012
1. Find the fundamental matrix (t, 0) associated with the linear system
3 2
2 3
a) A =
2.
,
1 4
1 1
b) A =
dx
dt
= Ax(t) where,
.
a) Find the fundamental matrix (t, t0 ) of the DE,
x (t) = A(t)x(t),
2t 0
1 2t
wi
AM 351: Assignment 3: Solutions
Due October 24th, 2012
1. Find the fundamental matrix (t, 0) associated with the linear system
3 2
2 3
a) A =
,
1 4
1 1
b) A =
dx
dt
= Ax(t) where,
.
SOLUTION.
a) The eigenvalues of this matrix are 1 = 5, = 1 with correspon
AM 351: Assignment 4 Revised
Due November 2, 2012
1. For the following nonlinear systems, identify all steady states of the system and classify their
dy
stability type. Then, use the fact that the ratio dx = dy / dx is separable to arrive at explicit
dt d
AM351 Spring 08
Due Friday, July 18 2008
Assignment 3
Instructions: write your solutions clearly, explain your steps carefully and refer to theorems and lemmas when needed. The assignment is due either in class or in the assignments drop box (4th f
AM351 Spring 08
Due Friday, June 20
Assignment 2
Instructions: write your solutions clearly, explain your steps carefully and refer to theorems and lemmas when needed. The assignment is due either in class or in the assignments drop box (4th floor)
AM351 Spring 08
Due Monday, June 9
Assignment 1
Instructions: write your solutions clearly, explain your steps carefully and refer to theorems and lemmas when needed. The assignment is due either in class or in the assignments drop box (4th floor)
AMATH 351
Assignment No. 1
Fall 2005
Warmup/Review/Introduction
For review and not to be handed in: AM351 Course Notes, Problem Set No. 1, Questions R1,R2,R3 (attached to this assignment). The following problems are due Wednesday, September 21, 2005. 1. S
AMATH 351
Due: Friday, September 30, 2005
Assignment No. 2
Fall 2005
1. Recall that in class, we studied the clamped string problem to arrive at the PDE, 2y T 2y = . t2 x2 (1)
Here y (x, t) denotes the displacement of the string from equilibrium. Using se
AMATH 351
Assignment No. 3
Fall 2005
Due: Friday, October 7, 2005, 1:30 p.m. (You can slide it under my door if I am not in my oce.) Unless otherwise indicated, all power series solutions are to be about the point x0 = 0. In each question, justify the typ
AMATH 351
Assignment No. 4
Fall 2005
Questions 1-5(a) are due on Thursday, October 20, 2005, 1:30 p.m. (You can slide the assignment under my door if I am not in my oce.) 1. In the Course Notes, p. 7-9, it is shown that any solution of the DE x2 dy d2 y +
AMATH 351
Assignment No. 5
Fall 2005
Reminder: Midterm examination, Thursday, November 3, 2005, 7:00-8:30 p.m. in MC 4045. You are responsible for all material in Lectures 1-18 (except the theoretical aspects of Picards method in Lecture 17), Assignments
AM 351
Assignment No. 6
Systems of First Order Linear DEs
Fall 2005
Due: Tuesday, November 21, 2005, 1:30 p.m. (under my door) 1. Construct the rst three iterates u 0 , u1 and u2 of Picards method of successive substitution applied to the following linear