Introduction to Partial Dierential Equations
John Douglas Moore
May 21, 2003
Preface
Partial dierential equations are often used to construct models of the most
basic theories underlying physics and engineering. For example, the system of
partial dierenti

ENGRD 221 Thermodynamics (Prof. N. Zabaras)
Prelim I
10/12/2006
Thursday October 12, 7:30 pm 9:30 pm Closed books and notes. Wireless or any other form of internet/phone connection is strongly prohibited during the exam. Answer all questions. Make sure yo

M244: Solutions to Final Exam Review
1. Solve (use any method if not otherwise specied):
(a) (2x 3x2 ) dx = t cos(t)
dt
SOLUTION: As a separable DE:
2x 3x2 dx =
t cos(t) dt x2 x3 = cos(t) + t sin(t) + C
You can leave your answer in implicit form.
(b) y +

ENGRD 221 Thermodynamics (Prof. N. Zabaras)
Prelim I
10/12/2006
Thursday October 12, 7:30 pm 9:30 pm
Closed books and notes.
Wireless or any other form of internet/phone connection is strongly prohibited during
the exam.
Answer all questions. Make sure yo

ENGRD 221 Thermodynamics (Prof. N. Zabaras)
Prelim 1
10/11/2007
Thursday October 11, 7:30 pm 9:30 pm
Closed books and notes. Answer all questions. Make sure your answers are legible.
Circle your final answer.
The TAs and instructor will not respond to any

ACM95b/100b Lecture Notes
Caltech 2004
The Method of Frobenius
Consider the equation
x2 y + xp(x)y + q(x)y = 0,
(1)
where x = 0 is a regular singular point. Then p(x) and q(x) are analytic at the origin and have convergent
power series expansions
pk x k ,

ENGRD 221 Prof. N. Zabaras
09/17/2007
Recitation Handout 4
09/17/2007 09/24/2007
KELVIN-PLANCK STATEMENT OF SECOND LAW
Page 1 of 6
ENGRD 221 Prof. N. Zabaras
For refrigerator:
09/17/2007
For heat pump:
A truly Reversible process is ONLY theoretical; a lis

09/17/2007
ENGRD 221 Prof. N. Zabaras
HOMEWORK 4
Handed out: Monday, 17 September 2007
Due: Monday, 24 September 2007 by 5 pm
Notes:
1. Drop HW off in the designated homework box in Upson 123
2. Be sure that you indicate your recitation section (and your

9/03/07
ENGRD 221 Prof. N. Zabaras
HOMEWORK 2
Handed out: Monday, 3 September 2007 Due: Monday, 10 September 2007, 5 pm
Notes:
1. Drop HW off in the designated homework box in Upson 123
2. Be sure that you indicate your recitation section (and your name!)

9/03/07
ENGRD 221 Prof. N. Zabaras
Recitation Handout 2
ENRGD 221: Engineering Thermodynamics Prof. Zabaras
Topics covered in class: Chapters 2 and 3
RETRIEVING THERMODYNAMIC PROPERTIES
The emphasis in this class is to retrieve thermodynamic properties us

ENGRD 221 Prof. N. Zabaras
09/10/2007
Recitation Handout 3
09/10/2007 09/16/2007
Part A: Recitation Notes
CONSERVATION OF MASS
Conservation of mass states that the time rate of
change of mass in a control volume given by
dmcv
= mi m e
dt
i
e
where mi and

ENGRD 221 Prof. N. Zabaras
09/24/2007
Recitation Handout 5
09/24/2007 10/1/2007
Topics covered in Class: Chapter 6 Using Entropy
1. Entropy Production:
cycle is defined as the entropy produced.
If cycle = 0; the process is reversible, there are no irreve

ENGRD 221 Prof. N. Zabaras
HOMEWORK 6 Handed out: Monday, 1 October 2007 Due: Friday, 5 October 2007 by 5 pm
10/1/07
Notes: 1. Drop HW off in the designated homework box in Upson 123 2. Be sure that you indicate your recitation section (and your name!) on

Faculty of Mechanical Engineering
ME 231: Thermodynamics I
Fall 2008
Instructor: Prof, Javed Ahmad Chattha
Room: G 01 FME
e-mail: chattha@giki.edu.pk
Lectures:
Tue - Thu, 8:00-8:50 AM
Main Lecture Hall FME
Textbook:
Fundamentals of Engineering Thermodynam

Ch 5.8: Bessels Equation
! Bessel Equation of order :
! Note that x = 0 is a regular singular point.
! Friedrich Wilhelm Bessel (1784 1846) studied disturbances
in planetary motion, which led him in 1824 to make the first
systematic analysis of solutions

09/24/2007
ENGRD 221 Prof. N. Zabaras
SOLUTION TO HOMEWORK 4
Problem 1
Given: Data are provided for a valve and turbine in series, each operating at steady state.
Find: For the turbine, determine the temperature at the inlet and the power developed per
un

9/17/2007
ENGRD 221 Prof. N. Zabaras
SOLUTIONS TO HOMEWORK 3
Problem 1
Known: Data are provided for a process of water contained in a piston cylinder assembly
Find: Determine the mass of water and Q for the process
Assumption:
1. the water in the piston-c

9/7/2007
ENGRD 221 Prof. N. Zabaras
SOLUTIONS TO HOMEWORK 2
Problem 1
Known: 5 kg of steam undergoes an expansion in a piston-cylinder assembly from state 1
to 2. During the process there is a known heat transfer to the steam and a known work
transfer of

Ch 5.7: Series Solutions
Near a Regular Singular Point, Part II
! Recall from Section 5.6 (Part I): The point x0 = 0 is a regular
singular point of
with
and corresponding Euler Equation
! We assume solutions have the form
Substitute Derivatives into ODE

Math 391 Problems with Answers
Professor Chavel
To the student: The following problems are lifted from my les. Some errors may have crept
into the text during the copy/paste. Some problems might be repeated. Also, note the approach
to second order equatio

2401 Dierential Equations: Bessel Functions
Page 1
Bessels Equation
There are many special functions which arise as solutions to dierential equations (Hermite, Legendre, Chebyshev, etc.).
Here we will look at how one important class of functions, Bessel F