642:527
ASSIGNMENT 1
FALL 2012
Turn in starred problems, and only those, Wednesday 09/12/2012.
Multiple-page homework must be STAPLED when handed in.
Section 4.2:
1 (a), (d), *(e), (g), *(k)
2 (b), (d), *(e)
7 (a), *(f), (i), (j)
Exercises from the Not

642:527
SOLUTIONS: ASSIGNMENT 10
FALL 2007
Section 17.3: 4 (g) f (x) = | sin x| has period (since sin(x + ) = sin x) so its Fourier series will have the form
FS f = a0 +
[an cos 2nx + bn sin 2nx].
n=1
Since f (x) is even (| sin(x)| = | sin x| = | sin x|)

642:527
SOLUTIONS: ASSIGNMENT 7
FALL 2007
NOTE: A. There is a Maple worksheet on the Assignments and solutions web page which shows the phase plane portraits for problems 2(b) and 2(k). B. Vectors are denoted either by boldface or, for Greek letters, unde

642:527
SOLUTIONS: ASSIGNMENT 5
FALL 2007
Some of these solutions were written by Professor Dan Ocone. 5.5.1 (a),(d): See solutions in text. (b) f (t) = H (t)et H (t 1)et . The rst H (t) is in a sense not needed, since in discussing the Laplace transform

642:527
SOLUTIONS: ASSIGNMENT 4
FALL 2007
Some of these solutions were written by Professor Dan Ocone. 5.2.1 A function f has exponential order as t if there exist constants K 0, T 0, and a constant c, such that |f (t)| Kect for all t T . (4.1)
A straight

642:527
SOLUTIONS: ASSIGNMENT 3
FALL 2007
Some of these solutions were written by Professor Dan Ocone. 4.5.8 The solutions in the text show how to do these. 9. In the denition (14) of the Gamma function we make the change of variable t = 2u2 , dt = 2u du,

642:527
SOLUTIONS: ASSIGNMENT 2
FALL 2007
Some of these solutions were written by Professor Dan Ocone. 4.3.6 (a) The equation is 2x2 y + xy + x4 y = 0. (We have multiplied through by an additional factor of x to simplify bookkeeping.) Substitution of 0 an

642:527
SOLUTIONS: ASSIGNMENT 1
FALL 2007
Some of these solutions were written by Prof. Dan Ocone. 4.2 Rather than using (7a) or (7b) it is usually better use the ratio test directly. (n + 1)xn+1 n+1 1. (a) Since lim = |x|, the series 0 nxn converges if =

642:527
FALL 2007
SUMMARY OF THE METHOD OF FROBENIUS Consider the linear, homogeneous, second order equation: y + p(x)y + q(x)y = 0. Suppose that x = 0 a regular singular point:
(1)
xp(x) =
n=0
pn x ,
n
|x| < R1 ,
x q(x) =
n=0
2
qn xn ,
|x| < R2 ,
R1 ,

s := N -> 1/2+sum(2*sin(2*k+1)*x)/(2*k+1)*Pi),k=0.N-1);
s(3);
plot([s(0),s(1),s(3),s(20),s(50)],x=0.2*Pi,color=[RED,GREEN, MAGENTA,BROWN,BLUE],thickness=1,numpoints=500);
b := n -> (10-n^2)/(n^4-16*n^2+100): a := n -> -2*n/(n^4-16* n^2+100): y := N -> 1/2

Competing Species Models
The general equations for competing species are of the form x = x(1 1x 1y) y = y(2 2y 2x),
where x and y are the populations of two species in some environment, 1 and 2 are positive constants representing the growth rates of these

Review problems: Math 527, Exam 2 Problems 16 are from last year's second midterm exam; problem 7 is additional. 1. Consider the system x = 1 - xy, y = x - y 3 . Determine its singular (equilibrium) points and classify each, insofar as possible, using lin

642:527 THE HEAT EQUATION IN A DISK Periodic and singular Sturm-Liouville problems
FALL 2007
In these notes we study the two-dimensional heat equation in a disk of radius a: 2 2 u(x, y, t) = u(x, y, t), t x2 + y 2 a 2 .
Here 2 , also written as , is the t

642:527
ASSIGNMENT 12: REVISED
FALL 2007
Multiple-page homework must be STAPLED when handed in. Turn in starred problems Thursday 12/6/2007. Problems marked with two stars will be treated as extra credit.problems Section 17.7: 8 Section 18.3: 10 (a), (b),

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w k x g q k gm j n " g m h fiifzPSor&5V gPeSieiP&0edicfw_ iSfP#iSePeergPePeTif h k g g q g q k l | m x g k h n gm l k j g q gm l k k j g n k h g m g q k g m h g q g m w ' w ' g j k m n k sm k j g h g g fiPfiPeSPtPeSi$doPo5yy0yC#0tTao0qP0ii21 k gm l m g

642:527
SOLUTIONS: ASSIGNMENT 8
FALL 2007
Section 7.4: 7 (a) By introducing y = x we can reformulate the given equation as x = y, y = 2P x x 1 ml l
2 1/2
k x. m
The right hand sides vanish when x = y = 0 so the origin is indeed a critical point. To linear

642:527
SOLUTIONS: ASSIGNMENT 11
FALL 2007
Section 17.7:1. In all sections of this problem the equation is y + y = 0; comparing with the standard Sturm-Liouville form (1.a) we see that p(x) = 1, q (x) = 0, and w(x) = 1. The solutions of the dierential equ

642:527
ASSIGNMENT 2
FALL 2012
Turn in starred problems, and only starred problems, Wednesday 09/19/2012.
Multiple-page homework must be STAPLED when handed in.
Section 4.3:
1 (a), (b), (c), *(g), *(l)
2
6 (a), *(e), *(p), *(t)
Hints and remarks: 1. In

642:527
ASSIGNMENT 3
FALL 2012
Turn in starred problems (including problem 3.A) on Wednesday 09/26/2012.
Do not turn in solutions for any unstarred problems.
Multiple-page homework must be STAPLED when handed in.
Section 4.3:
6 (f), (u), *(v)
Section 4.5

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ASSIGNMENT 4
FALL 2012
EXAM 1 IS SCHEDULED FOR WEDNESDAY, OCTOBER 10.
Turn in starred problems on Wednesday 10/05/2012. Do not turn in unstarred
problems. Multiple-page homework must be stapled.
Section 5.2: 1 (a), (c), (g), (j); 5; 6; 8; 11
Secti

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ASSIGNMENT 5
FALL 2012
EXAM 1 IS SCHEDULED FOR WEDNESDAY, OCTOBER 10. The exam
will cover all our work on series solutions of ordinary dierential equations (including
basic properties of p[ower series, solutions at ordinary and regular singular po

642:527
ASSIGNMENT 7
FALL 2012
Turn in starred problems Wednesday 10/24/2012.
Section 7.3: 11 (a), (d)*
Section 7.4: 2 (a), (b)*, (e), (f)* See instructions below.
Problem 7.A* Consider the equations
x = x + 3y + ax(x2 + y 2 ),
y = x y + ay (x2 + y 2 ).
I

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ASSIGNMENT 8
FALL 2012
Due to the class cancellation on Monday and Wednesday, October 29 and 31, we are
returning to the original Assignment 8. The problems from Chapter 9, and problem
8.B, have beem restored, and the assignment is now due Wednesd

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ASSIGNMENT 9
FALL 2012
This is a brief and relatively easy assignment; we will cover the necessary material
(Sections 17.2 and 17.3, plus Chapter 1 and Section 2.1 of the posted notes) on or before
Wednesday 11/7. Assignment 10 will be made early;

642:527
ASSIGNMENT 10
FALL 2012
The second exam will be on Monday, November 19. A description of the coverage
on the exam, and a set of review problems, is posted on the web page. We will have a
review/problem session on Friday, November 16, 1:403:00 PM,

642:527
ASSIGNMENT 12
FALL 2012
Turn in starred problems Monday 12/10/2012.
Section 18.3: 6 (f)*, (i), (l), (m), 10 (a), (c)*, (e), (f)*, 14, 15, 19*, 29
13.A* Here is a variant of the periodic boundary condition problem of Section 17.8:
Find the eigenval

642:527
ASSIGNMENT 13
FALL 2012
No problems from this assignment will be collected.
Problems on the Fourier transform and the wave equation:
Section 17.10: 2, 3*, 4 (c), (f), 6 (a), (c), (g)
Section 18.4: 1, 6, 8(a,b)
Section 19.2: 2(a,bc), 5, 8
Section 1