21-260: Dierential Equations
Wave Equation Examples
Wave equation:
2 uxx = utt ,
0 < x < L,
t > 0;
u(0, t) = 0,
u(L, t) = 0,
t 0;
u(x, 0) = f (x),
ut (x, 0) = g(x),
0 x L.
Solution:
u(x, t) =
cn sin
n=1
nx
cos
L
Here cn is the coecient of sin
nx
L
2
cn =
Math 260: Dierential Equations
Nonhomogeneous Equations
Method of Undetermined Coe cients
Consider the equation
ay 00 + by 0 + cy = g(t).
1. Let g(t) = Pn (t) = a0 tn + a1 tn 1 + . . . + an . Let s be the algebraic multiplicity of
a 2 + b + c = 0. Then nd
Last Name:
First Name:
Dierential Equations Final Exam Practice
Please practice circling your section!
Devi Borg
8:30 a.m.
Section A
Antoine Remond
9:30 a.m.
Section E
Devi Borg
9:30 a.m.
Section B
Devi Borg
10:30 a.m.
Section C
Antoine Remond
11:30 a.m.
Dierential Equations
Homework Assignment 9 Brief Answers
Disclaimer: This le contains only the nal answers, not the full solutions to the homework. Students are expected to show all the relevant work in their homework assignments.
Section 7.2 Problem 30.
Dierential Equations
Homework Assignment 8 Brief Answers
Disclaimer: This le contains only the nal answers, not the full solutions to the homework. Students are expected to show all the relevant work in their homework assignments.
Section 5.1 Problems 7.
Dierential Equations
Homework Assignment 4 Brief Answers
Disclaimer: This le contains only the nal answers, not the full solutions to the homework. Students are expected to show all the relevant work in their homework assignments.
Problem 1. Not graded.
21-260: Dierential Equations
Final Exam Formula Sheet
1. If M (x, y)dx + N (x, y)dy = 0 is not exact, you might be able to nd and integrating
factor that will make it exact. If (My Nx )/N is a function of x only, then there is an
integrating factor (x) th
Dierential Equations
Homework Assignment 3 Brief Answers
Disclaimer: This le contains only the nal answers, not the full solutions to the homework. Students are expected to show all the relevant work in their homework assignments.
Problem A. Graded out of
Dierential Equations
Homework Assignment 1 Brief Answers
Disclaimer: This le contains only the nal answers, not the full solutions to the homework. Students are expected to show all the relevant work in their homework assignments.
Problem A. Graded out of
Dierential Equations
Homework Assignment 7 Brief Answers
Disclaimer: This le contains only the nal answers, not the full solutions to the homework. Students are expected to show all the relevant work in their homework assignments.
Section 3.3 Problem 6. G
Dierential Equations
Homework Assignment 6 Brief Answers
Disclaimer: This le contains only the nal answers, not the full solutions to the homework. Students are expected to show all the relevant work in their homework assignments.
1. (a)
W [x(1) , x(2) ]
Dierential Equations
Homework Assignment 10 Brief Answers
Disclaimer: This le contains only the nal answers, not the full solutions to the homework. Students are expected to show all the relevant work in their homework assignments.
Problem 2. Sketch the g
Dierential Equations Final Exam Practice
Solutions
1. A tank originally contains 10 gal of water with 1/2 lb of salt in solution.
1
Water containing a salt concentration of 200 (10 t)2 (sin(t) + 1) lb per gallon
ows into the tank at a rate of 1 gal/ min,
Homework 9
Sidharth Madan
[email protected]
C
November 17, 2011
A17
7.47
7.11
Equivalence relations R on a set S.
a) S= N - 1; (x,y) R if and only if x and y have a common factor bigger than 1.
Thus, we can see that this Relation is reexive as (x,x)
21-127: Concepts of Mathematics
Homework 10 (due Tuesday, December 06)
Directions: Write up carefully argued solutions to the following problems. The rst task is to be complete and correct. The more subtle task is to keep it simple and succinct. Your solu
21-127: Concepts of Mathematics
Homework 4 Solutions
A9 (10 points)
Determine the set of natural numbers that can be expressed as the sum of some non-negative number
of 3s and 5s. That is, the set of numbers, S , where
S = cfw_3k + 5j | k, j N cfw_0
Solut
F ORMAL L ANGUAGES , AUTOMATA AND
C OMPUTATION
NP-C OMPLETENESS
Carnegie Mellon University in Qatar
( L ECTURE 20)
S LIDES FOR 15-453
S PRING 2011
1 / 30
S UMMARY
Time complexity: Big-O notation, asympotic complexity
Simulation of multi-tape TMs with a si
F ORMAL L ANGUAGES , AUTOMATA AND
C OMPUTATION
C OMPLEXITY
( L ECTURE 19)
S LIDES FOR 15-453
S PRING 2011
1 / 41
C OMPLEXITY T HEORY
Q UESTION
Assume that a problem (language) is decidable. Does that mean we
can realistically solve it?
A NSWER
NO, not alw
F ORMAL L ANGUAGES , AUTOMATA AND
C OMPUTATION
A DVANCED T OPICS IN C OMPUTABILITY
( L ECTURE 18)
S LIDES FOR 15-453
S PRING 2011
1 / 23
R ICE S T HEOREM M OTIVATION
Consider the following undecidable languages:
ETM = cfw_ M | M is a TM and L(M ) =
TOTAL
F ORMAL L ANGUAGES , AUTOMATA AND
C OMPUTATION
P OST C ORRESPONDENCE P ROBLEM
( Lecture 17)
Slides for 15-453
Spring 2011
1 / 28
R EVIEW OF D ECIDABILITY AND R EDUCTIONS
( Lecture 17)
Slides for 15-453
Spring 2011
2 / 28
R EDUCIBILITY
A reduction is a way
F ORMAL L ANGUAGES , AUTOMATA AND
C OMPUTATION
R EDUCIBILITY
( L ECTURE 16)
S LIDES FOR 15-453
S PRING 2011
1 / 20
T HE L ANDSCAPE OF THE C HOMSKY H IERARCHY
( L ECTURE 16)
S LIDES FOR 15-453
S PRING 2011
2 / 20
R EDUCIBILITY
A reduction is a way of conve
F ORMAL L ANGUAGES , AUTOMATA AND
C OMPUTATION
D ECIDABILITY
( L ECTURE 15)
S LIDES FOR 15-453
S PRING 2011
1 / 34
T URING M ACHINES -S YNOPSIS
The most general model of computation
Computations of a TM are described by a sequence of
congurations. (Accept
F ORMAL L ANGUAGES , AUTOMATA AND
C OMPUTATION
T URING M ACHINES
Carnegie Mellon University in Qatar
( L ECTURE 14)
S LIDES FOR 15-453
S PRING 2011
1 / 30
T URING M ACHINES -S YNOPSIS
The most general model of computation
Computations of a TM are describe
F ORMAL L ANGUAGES , AUTOMATA AND
C OMPUTATION
T URING M ACHINES
Carnegie Mellon University in Qatar
( L ECTURE 13)
S LIDES FOR 15-453
S PRING 2011
1 / 32
T URING M ACHINES -S YNOPSIS
The most general model of computation
Computations of a TM are describe
21-260: Dierential Equations
Fourier Series by Visual Inspection
1. Write the Fourier series for f (x) = 2 cos (x) + sin (3x) + 7 cos (4x) on [, ].
Solution. Notice that L = , thus the Fourier series of f (x) will have the form
a0
+
[an cos (nx) + bn sin