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
e2x y =
x 2x 1 2x
e e + C.
2
4
Solve for y and substit
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 polynomial cA () =
det(A I ) is given.
(b) Find all of the ei
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 = 1. From Theorem 7.1, we need to solve the system of
die
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 computation of the inverse A1 of a square
matrix. The two most
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
e2x y =
x 2x 1 2x
e e + C.
2
4
Solve for y and substit
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
6 and 8 of Chapter 2 in Waltman. You will be allowed (
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) = 5te2t e2t
(b) y (t) = 2e3t cos 2t + e3t
(c) y (t) = t4
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 can be adequately assessed. Credit will not be given f
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 correct ones) without supporting work. Put your name on each
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, 0). Which two systems have the same
qualitative struct
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) and (1
2, 1)
(e) (0, 0), (1, 1), and (1, 1)
(f) cfw_(
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() = 0 are 1 = 0 and 2 = 1/4 so the solution of the die
Name:
Exam 2
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 correct ones) without supporting work. Put your name on each
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 x 2x 1 2x e e + C. 2 4 Solve for y and substitute y (0
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) = ex+2 Solution. These functions are linearly dependent
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 indicated. (a) y 2x2 y = 0 (b) y 4xy + (4x2 2)y = 0 (c) (1 x
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 + 1)(r 1) has roots 1 and 1/2. Hence y = c1 x + c2 |x|
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 = x2 dx and integrate to get an implicit equation y2 x