Differential Equations
LECTURE 4O
Separation of Variables and Heat Equation IVPS
1. Initial Value Problems
Partial differential equations generally have lots of solutions. To specify a unique one, well
need some additional conditions. These conditions are
2015 Math 4265/6265 Graded Homework 3 Due: Tuesday, Dec. 1
Maximum: 20 pts (16 pts if returned after the deadline)
Name:
1. 1. Find the Fourier transform
f ( x)
-1 if
1 if
0
-1 x 0
0 x 1
otherwise
.
2. Represent f(x,y) by a double Fourier series, where 0<
Math 4265/6265 Fall 2015 - Study Guide - Test III
Review:
Sections 11.8,11.9 (Fourier integral, Fourier transform), 12.6 (Steady 2D
Problems),12.7,12.8,12.9
Review all homework assignments.
You should also know how to derive the solution of the heat equat
2015 Fall, Math 4265/6265 Graded Homework 1
Due: Tuesday, September 15
Maximum: 20 pts (16 pts if returned after the deadline)
Name:
.
10 pts 1. Solve the given ODE by the power series method
y xy y 0
10 pts 2. Solve the given ODE by the Frobenius method
2015 Fall, Math 4265/6265 Graded Homework 2
Due: Tuesday, October 27
Maximum: 20 pts (16 pts if returned after the deadline)
Name:
.
10 pts Find the series solution to the wave equation utt 4u xx , 0 x 6,
subject to the boundary conditions u(0, t ) 0, u(6
Math 4265/6265 Fall 2015 - Study Guide - Test I
Review: Sections 5.1-5.3, 11.1-11.2, and 12.1 (all Homework assignments)
Practice test:
1. Solve the given ODE by the power series method
( x5 4 x3 ) y (5x 4 12 x 2 ) y
2. Solve the initial value problem for
Math 4265/6265 Fall 2015 - Study Guide - Test II
Review: Sections 12.2-12.6 [up to Steady Two Dimensional Heat Problems], including
the HW assignments. You may use a one page formula sheet.
Practice test:
1. Find the series solution to the wave equation u
Constant Coefficients
y=c 1 e
Distinct Real:
m1 x
y=c 1 e
Repeated Real:
+ c2 e
m1 x
1
cosnx cosmx= cfw_cos ( n+ m ) x +cos ( nm) x
2
m2 x
m1 x
+ c2 x e
;
x
.
x
y=e (c 1 cos x+ c 2 sin x)
Complex Conjugate:
1
sin nx cos mx= cfw_sin ( n+m ) x +sin ( nm)
Constant Coefficients
y=c 1 e
Distinct Real:
m1 x
y=c 1 e
Repeated Real:
+ c2 e
m1 x
m2 x
m1 x
+ c2 x e
;
x
.
x
Complex Conjugate:
y=e (c 1 cos x+ c 2 sin x)
Wave Equation with
g( x)=0
u ( x , t )= Bn cos n t sin
n=1
u ( x , 0 )= B n sin
n=1
n
c n
x n=
g( x)=0
Wave Equation with
u ( x , t )= Bn cos n t sin
n=1
u ( x , y )= A sinh
n
n=1
n
x
L
n=1
n
u ( x , 0 )= B n sin
x=f (x )
L
n=1
L
A n=
Heat Equation with Ends Kept at Temperature
0
u ( x , t )= Bn sin
n=1
n t
xe
L
u ( x , 0 )= B n sin
n=1
2
n
n
x
L
Question 1
Your answer is CORRECT.
Calculate the given integral
I IS 3 xi
9 dz
gr 83: +6:
Question 2
Your answer is CORRECT.
Suppose x) is an invertible differentiable functian and
R31): 1. In): 5, f (-3}: -5-, f (=3
Evaluate the derivative of the inverse
1 . (15 points) Evaluate the line integral / ($2 + 3/2) d5 along the path given by the vector
C
function 'r-(t) = [cos(t) + tsin(t)] 73+ [sin(t) tcos(t)] j, for t E [0, 2].
.2.
8mm: WW3" 1:} rm) mm o1:
.I
Him) = [mm +6 CA»an + [yam wt w/f)12
(942098 +
1. , Suppose that Q(a:, y, z) is the statement 1122+y2 = 22. What are the truth values of
the three statements below when the universe of discourse is the set of all integers?
2 /r/
r (a) 3nyQ(éc,-y,x) 2, a,
. f ,x-:~xé Q\
i (b) 321V$Q(§;,y,556)(3>dp+ Y2
Sec 8.2 Series
Part 1
A series is simply the sum of the terms of a sequence.
2 arr1 2611 + a2 + a3 + a4 + - - is called an innite series where an is the nth term of the series.
11:1
Associated With any innite series is a sequence (list), {sn} = s1, 52, S3
. Work on the Quiz# 7 and last classworkithat I shall give in last class.
. See the true false and concept questions at the end of chapter 12 16.
. Take a look of previous exams.
. Read the relevant sections, mainly the concepts in the boxes and the ones
Answer each question completely. Show all work. No credit is allowed for
mere answers with no work shown. Show the steps of calculations. State the
reasons that justify conclusions.
1. (12 points) Find parametric equations of the line of intersection of
1 UL! llluSL SHOW an WUI'K 1621(1ng L0 [116 11113.! answer.
Work only on these pages, using methods used in class. Credit will not be given for any answer that is not supported.
No calculators.
so _.
You are expected todo basic calculations to simplify, f
Show all your work when requested .to receive credit. You may not use a calculator on this test. If you
use a specic test to show convergence or divergence of a series, you must state the test you use when it is
requested. Clearly label your answers.
1. (
ID # Class Time or CRN
READ THE DIRECTIONS. YOU MUST SHOW ALL WORK ON THIS TEST AND USE METHODS
LEARNED IN CLASS TO RECEIVE FULL CREDIT. YOU MAY USE THE CALCULATORS
DISTRIBUTED IN CLASS. (100 pt.)
1. Differentiate the following. W (35 pt)
3. y=5x4+l+x"+e+
Instructions: Please enter your NAME, ID Number, FORM DESIGNATION, and your CRN on the op-
scan sheet. The CRN should be written in the box labeled COURSE. Do not include the course number.
Darken the appropriate circles below your ID number and below the
1 : a) (20 pts.) Find the plane that contains the line
7m y3 25
and the point with coordinates P(2, 4, 6).
b) ( 5 pts.) Find the line that passes through the origin and is orthogonal to the plane you found
in part (a).
2 : a) (15 pts.) Consider the
Use no books, notes, or calculators. Show all your work.
QUESTION 1 (30 points): Consider the initial value problem
11" + 23f + 53; = 0, y(0) = 2, y(0) = 2.
(a) Solve this initial value problem_
Exo'mpe Q , p . 36
-t' , -. -\
g m = 9,9 (was a: * 511-)
(b)
COURSE SYLLABUS
MATH 4265/6265
Partial Differential Equations
Fall 2015
Days & Time: TR 02:30 pm - 03:45 pm
Room:
Langdale Hall # 323
CRN:
82614
Instructor:
Dr. Igor Belykh
Office:
College of Education Building, # 784
Office Hours: TR 1:30-2:30 pm, other