Exercise Set 2
Math 4027
Due: January 25, 2007
1. Find the solution of the initial value problems:
(a) y + 2y = x,
y (0) = 1,
Solution. Multiply by e2x to get (e2x y ) = xe2x and then integrate to get
Exercise Set 4
Math 4027
Due: February 15, 2007
For each of the following matrices A, do all of the following calculations:
(a) Compute the eigenvalues of A. For convenience, the characteristic polyno
Exercise Set 5
Math 4027
Due: March 1, 2007
From Waltman, Section 7.
1. Find eAt where
(a) A =
11
.
01
Solution. The matrix A is triangular, so the eigenvalues are the diagonal
entries, i.e., 1 = 2 =
Exercise Set 6
Math 4027
Due: March 21, 2007
Find the general solution of the given dierential equation.
1. 4y + y = 0
Solution. The characteristic polynomial is p() = 42 + = (4 + 1) so the roots
of p
Exercise Set 3
Math 4027
Due: February 6, 2007
Pages 1314.
7. Construct the inverse of each of the given matrices. You may use any of the techniques
that you learned in linear algebra for the computat
Exercise Set 2
Math 4027
Due: January 25, 2007
1. Find the solution of the initial value problems:
(a) y + 2y = x,
y (0) = 1,
Solution. Multiply by e2x to get (e2x y ) = xe2x and then integrate to get
Math 4027 Exam 2 Review Sheet
Exam 2 will be on Tuesday, April 24, 2007. The syllabus for this exam consists of
Sections 9 (Elementary Stability) and 11 (Scalar Equations) of Chapter 1 and Sections 1
Math 4027 Exam 1 Review Sheet
Review Exercises for Exam 1
Answers
1. (a) y (t) = ce2t
(b) y (t) = ce2t + et
(c) y (t) = cet + 1 e3t
2
2
(d) y (t) = cet +
(e) y (t) = ct3 t
1
2
3
cos t
2. (a) y (t) = 5
Name:
Exam 2
Instructions. Answer each of the questions on your own paper, except for problem 2,
where you may record your answers in the box provided. Be sure to show your work so that
partial credit
Name:
Exam 1
Instructions. Answer each of the questions on your own paper. Be sure to show your
work so that partial credit can be adequately assessed. Credit will not be given for answers
(even corre
Exercise Set 8
Math 4027
Due: April 19, 2007
1. Consider the three systems
(a)
x
y
= 2x + y
= y + x2
(b)
x
y
= 2x + y
= y + x2
(c)
x
y
= 2x + y
= y x2
All three have a critical point at the origin (0,
Exercise Set 7
Math 4027
Due: April 10, 2007
From Waltman, Page 107.
6. Locate the critical points of the following systems.
(a) cfw_(n, 0) : n Z
(b) (0, 0) and (1, 1)
(c) cfw_(0, y ) : y R
(d) (0, 0)
Exercise Set 1
Math 4027
Due: February 1, 2005
1. Find the solution of the initial value problems: (a) y + 2y = x, y (0) = 1,
Solution. Multiply by e2x to get (e2x y ) = xe2x and then integrate to get
Exercise Set 2
Math 4027
Due: February 10, 2005
1. Determine, with justication, whether each of the following lists of functions is linearly dependent or linearly independent. (a) 1 (x) = ex , 2 (x) =
Exercise Set 3
Math 4027
Due: March 15, 2005
1. Verify that the function 1 (x) is a solution of the given dierential equation, and nd a second linearly independent solution 2 (x) on the interval indic
Exercise Set 4
Math 4027
Due: April 7, 2005
1. Find the general solution of each of the following dierential equations. (a) 2x2 y + xy y = 0 Solution. The indicial equation q (r) = 2r(r 1) + r 1 = (2r
Exercise Set 5
Math 4027
Due: May 5, 2005
1. Solve the following dierential equations: (a) y = x2 /y . Solution. The equation is separable, so rewrite it in the form yy = x2 or in dierential form y dy
Final Exam Practice Problems
Math 4027
The nal exam will be comprehensive. You should review both the statements of all existence and uniqueness theorems and the various techniques employed for nding