Possibly helpful formulas and models
1
du = ln |u| + C
u
du
= arctan u + C
1 + u2
eu du = eu + C
un1 sin u du
un cos u du = un sin u n
nu
nu
u e du = u e n
n 1 u
u
un eu du = n!
e du
0
Equation of motion: x(t) is position at time t, a is acceleration (con
Page 1
1. (5) The following picture is a slope eld for the differential equation % = a: + y. Suppose y(:z:)
is a solution to the initial value problem 73 = :1: + y, y(0) = 2. Without nding y explicitly, what
can you say about y(250)? What about y(250)?
70
Math 3113 homework
1. (assigned 1/19) Study the course syllabus. Make sure you know the meaning of the
following terms: equation, dierential equation, ordinary dierential equation, partial
dierential equation, order of a DE, one-parameter family. Read som
Math 3113 homework
29. (assigned 3/9) Prepare for Quiz 5 on Friday. Over the spring break, do 3.5 # 47. Try
to do it (i) rst by the single formula in Theorem 1 of Section 3.5, then (ii) from the
two equations we called (1) and (2) in class, that is, formu
Math 3113 homework
38. (assigned 4/11) 5.2 # 2, 3, 4 (do not do the part of the problem that asks you to use
a computer system or graphing calculator, just use the eigenvalue method to nd the
general solution).
39. (assigned 4/13) 5.2 # 8, 9 (again, do no
Mathematics 3113-005
Name (please print)
Quiz 1 Form A
January 28, 2011
Instructions: Give concise answers, but clearly indicate your reasoning. Most of the problems have rather short
answers, so if you nd yourself involved in a lengthy calculation, it mi
Mathematics 3113-005
Name (please print)
Quiz 1 Form A
January 28, 2011
Instructions: Give concise answers, but clearly indicate your reasoning. Most of the problems have rather short
answers, so if you nd yourself involved in a lengthy calculation, it mi
Mathematics 3113-005
Name (please print)
Quiz 1 Form B
January 28, 2011
Instructions: Give concise answers, but clearly indicate your reasoning. Most of the problems have rather short
answers, so if you nd yourself involved in a lengthy calculation, it mi
Mathematics 3113-005
Name (please print)
Quiz 1 Form B
January 28, 2011
Instructions: Give concise answers, but clearly indicate your reasoning. Most of the problems have rather short
answers, so if you nd yourself involved in a lengthy calculation, it mi
Math 3113 homework
13. (assigned 2/11) Study the posted Quiz 2 solutions. The class performance on Quiz 2
was generally not what it should be, partly due to not following the instructions of the
problems. And many people even missed the repeat question fr
Mathematics 3113-005
Name (please print)
Quiz 2 Form B
February 11, 2011
Instructions: Give concise answers, but clearly indicate your reasoning.
I.
(3)
For the rst-order linear homogeneous DE y + P (x)y = 0, verify that if y1 and y2 are solutions, then s
Mathematics 3113-005
Name (please print)
Quiz 2 Form A
February 11, 2011
Instructions: Give concise answers, but clearly indicate your reasoning.
I.
(3)
dy
Check whether the initial value problem
= y 1/3 , y(3) = 0 satises the hypotheses of the Existence
Math 3113 homework
22. (assigned 2/25) Check the posted solutions of Quiz 3 to see what you didnt know
about the theory. This will help you determine what gaps you have in your knowledge
of sections 3.1 and 3.2. Get fully caught up on those sections, with
Mathematics 3113-005
Name (please print)
Quiz 2 Form A
February 11, 2011
Instructions: Give concise answers, but clearly indicate your reasoning.
I.
(3)
dy
Check whether the initial value problem
= y 1/3 , y(3) = 0 satises the hypotheses of the Existence
Mathematics 3113-005
Name (please print)
Quiz 2 Form B
February 11, 2011
Instructions: Give concise answers, but clearly indicate your reasoning.
I.
(3)
For the rst-order linear homogeneous DE y + P (x)y = 0, verify that if y1 and y2 are solutions, then s
Mathematics 3113-005
Name (please print)
Quiz 3 Form A
February 25, 2011
Instructions: Give concise answers, but clearly indicate your reasoning.
I.
Two linearly independent solutions of the DE y + 3y + 2y = 0 are ex and e2x (do not check these).
(6)
(a)
D
IFFERENTIAL
E
QUATIONS
Erin P. J. Pearse
These notes follow Dierential Equations and Boundary Value Problems (4ed) by C. H.
Edwards and D. E. Penney, but also include material borrowed freely from other sources.
This document is not to be used for any c
Linear Independence & Dierential Equations
Denition 1. A set of functions cfw_y1 , . . . , yn , yi : I R, is linearly independent i
you cannot write one function as a linear combination of the others. More precisely, i
c1 y 1 + + cn y n = 0
=
c1 = c2 = =
Partial Fractions Review
Introduction. The goal of partial fraction decomposition is to rewrite a rational function in a simpler form: split it into a sum of smaller rational functions where each denominator is an irreducible
polynomial. Here, irreducib
Practice Exam 1a
1. Solve the IVP:
2. Solve the ODE:
3. Solve the ODE:
dy
= y , y (0) = 2.
dx
dy
1+ x
=
.
dx
1+ y
(1 + ex )
2(ey + x3 e2x )dx xey dy = 0.
4. 100L of a brine solution is diluted with pure water at a rate of 2L/min and drained
at a rate of 1
MATH 3113, HOMEWORK 1, SOLUTIONS, DUE THURSDAY
JANUARY 23
1. Sketch the set of points z C determined by the condition
|z 1 + i| = 1.
This is the circle with radius 1 centered at 1 i.
2. Prove that for z C \ cfw_1,
1 + z + z2 + + zn =
1 z n+1
1z
(n = 1, 2,
MATH 3113, EXAM 1, SOLUTIONS, FEBRUARY 17
1. Find the general solution of the equation
xy 3y x3 = 0.
This is a rst order linear equation:
y 3x1 y = x2 .
We proceed as follows:
3x1 dx = 3ln(|x|),
(x) = |x|3 ,
|x|3 x2 dx =
y(x) = |x|3 (
|x|1 dx =
x
ln(|x|)
MATH 3113, EXAM 2, STUDY GUIDE
Exam 2 will cover the following sections: 1.3, Appendix, 8.1 and 8.2. Please read them
carefully and practice by working on the following problems. (Some of them have been
assigned previously as homework.)
Section 1.3, page