Dr. Z.s Calc4 Lecture 4 Handout: Existence and Uniqueness of First-Order Diff.Eqs.
By Doron Zeilberger
Version of Sept. 19, 2016 (correcting an omission pointed out by Emily Davis)
There are two important theorems that tells when you are guaranteed to hav

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lx/EATH 244 (13), Dr. Z. , FINAL EXAM, Friday, Dec. 20, 2013, 4:007:00pm,
PHllS .
NAME: (print!) MUQ
EMail address: ME? \L. 1:13
No Calculators! No Cheatsheets! Write the lial answer to each problem

CALCULUS BC
WORKSHEET 1 ON LOGISTIC GROWTH
NAME _
Do not use your calculator.
1. Suppose the population of bears in a national park grows according to the logistic differential
dP
5 P 0.002 P 2 , where P is the number of bears at time t in years.
equatio

Math 244 Exam 2 Practice Problems
1. Consider the initial value problem
( )
( )
(a). Find the solution.
( )
(b). Change the second initial condition to
and find
the solution as a function of
Then find the critical value
of that separates solutions that al

MATH244
FINAL EXAM Practice Problems
SPRING 2014
These are just practice problems. The final exam will look
different and may contain different material. Please see the
review document posted in sakai for a guide to final exam format
and contents.
1. Find

MATH 244
Spring 2014
Exam 1
02/26/2014
Time: 80 Minutes
Name:
Section
This exam contains 8 pages (including this cover page) and 7 problems. Check to see if any
pages are missing. Enter all requested information on the top of this page, and put your
initi

Math 244 Exam 1 Review
Material Covered: 1.1 to 1.3, 2.1 to 2.6, 3.1 and 3.2 (Gravity problems, 2.7, 8.1-3 omitted).
Format of the Exam and Material to Review:
Seven problems: First one is a non-computational one to test your overall understanding and is

Dr. Z.s Calc4 Lecture 8 Handout:
Homogeneous Second-Order Differential Equations With Constant Coefficients
By Doron Zeilberger
The general form of a general (usually non-linear) second-order diff.eq. is
y 00 (t) = F (t, y(t), y 0 (t)
.
Most diff. eq. can

Dr. Z.s Calc4 Lecture 7 Handout: Numerical Solutions of Ordinary Differential Equations
By Doron Zeilberger
Important Method: (Eulers method for solving a first-order ode)
For the initial value problem
y 0 = f (x, y) ,
y(x0 ) = y0
,
with mesh-size h, you

Dr. Z.s Calc4 Lecture 3 Handout: Separable Differential Equations
By Doron Zeilberger
A general first-order diff.eq. has the form
dy
= f (x, y) ,
dx
where f (x, y) is a function of both x and y, and x and y are intertwined. If you are extremely
lucky, the

Dr. Z.s Calc4 Lecture 2 Handout: Method of Integrating Factors for First-Order Linear Equations
By Doron Zeilberger
(Version of Sept. 6, 2013, correcting a typo found by Meredith Taghon )
Theory: After possibly dividing the coefficient of y 0 (t), any fir

Answers to Dr. Z.s Math 244 (Calc4) Homework assignments (when applicable)
Disclaimer: not responsible for any errors. The first finder of any error will get $1.
Version of Sept. 21, 2016: Thanks to Vinay Chang (correcting a typos in #4, problem 2).
Previ

MATH 244 (1-3), Dr. Z. , Solutions to Exam I, Mon., Oct. 14, 2013, 8:4010:00am, PH-115
Version of Oct. 16, 2013, 9:55am (thanks to Amar Patel, correctiong a typo in problem
6 (in the answer box)
1. (10 pts.) Find the general solution to the following diff

MATH 244 (1-3), Solutions to Dr. Z. , Exam II, Tue., Nov. 26, 2013, 8:4010:00am, PH-115
1. (10 pts.) Solve the initial value problem
4 y(t) + y 00 (t) = 3 sin t
y 0 (0) = 3 y(0) = 1.
,
Ans.:
y(t) = cos 2t + sin 2t + sin t
,
We must first rewrite it in the

Dr. Z.s Calc4 Lecture 1 Handout: Introducing Differential Equations
By Doron Zeilberger
Section 1: Direction Fields
A general first order differential equation looks like
dy
= f (y, t) ,
dt
where f (t, y) is (usually) a function of both t (usually time),

Dr. Z.s Calc4 Lecture 5 Handout: Autonomous Equation
By Doron Zeilberger
An autonomous first-order diff.eq. is a special case of a separable equation that has the form
y 0 (t) = f (y) ,
where the function on the right side only depends on y. Of course we

Dr. Z.s Calc4 Lecture 6 Handout
By Doron Zeilberger
[Version of Sept. 23, 2013, correcting a typo, thanks to Jonathan Chang]
Some first order diff.eqs. are neither linear nor separable, but you can still solve them exactly, at
least implicitly, (i.e. find

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MATH 244 Name: AMWU
Spring 2014
Exam 2
04/09/2014
Time: 80 lVIinutes _ _ ' Section
This exam contains 8 pages (including this cover page) and 8 problems. Check to see if any
pages are missing. Enter all requested information on the top of this page,