Math 257/316 Section 202 ' Midterm 2 March 13
Name: Student 7%:
2 questions; 50 minutes; max = 30 points;
Instructions: no calculators, books, notes, or electronics. Show and explain all your work.
1
Lecture 10: Fourier Sine Series
In the last lecture we reduced the problem of solving the initial-boundary value problem for the heat distribution along a
conducting rod to solving two ODEs, one in
Math 257/316 Section 202
2 questions;
Midterm 1
50 minutes;
February 6
max = 30 points
1. Consider this second order, linear, homogeneous ODE:
2(x
1)y 00 + y 0 + y = 0.
(a) Find the general solution i
Math 257/316 Section 202
Name:
Midterm 2
Student #:
March 13
2 questions;
50 minutes;
max = 30 points;
Instructions: no calculators, books, notes, or electronics. Show and explain all your work.
1. (1
1
Lecture 7: Introduction to Partial Dierential Equations
(continued)
In this lecture we will continue with the derivation of the basic PDEs studied in this course from dierent physical
situations. We
1
Lecture 8: Solving the Heat, Laplace and Wave equations
using nite dierence methods
In this lecture we introduce the nite dierence method that is widely used for approximating PDEs using the compute
1
Lecture 9: Separation of Variables and Fourier Series
In this lecture we will introduce the method of separation of variables by using it to solve the heat equation, which
reduces the solution of th
1
Lecture 6: Introduction to Partial Dierential Equations
In this lecture we will introduce the three basic partial dierential equations we consider in this course. After briey
discussing the classica
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Lecture 5: Examples of Frobenius Series: Bessels Equation
and Bessel Functions
In this lecture we will consider the Frobenius series solution of the Bessel equation, which arises during the process
Math 257/316 Assignment 8
Due Mon. Mar. 23 in class
1. Consider the wave equation utt = c2 uxx , 9 1 < x < 1, with initial position
8
< x + 1 if 1 < x 0 =
1 x if 0 < x < 1
u(x, 0) = f (x) =
and with i
1
Lecture 3: Regular Singular points
In this lecture we will dene a Regular Singular Point about which a Taylor series will not work. We will also introduce
the concept of the radius of convergence of
1
Lecture 2: Series solutions to ODE with variable coecients
In this lecture we will introduce series methods for the solution of variable coecient ODE. We introduce the concepts of
ordinary points ab
1
Lecture 4: Frobenius Series about Regular Singular Points
In this lecture we will summarize the classication of expansion points x0 for series as ordinary points for which Taylor
Series approximatio
1
Lecture 1: Review of methods to solve Ordinary
Dierential Equations
In this lecture we will briey review some of the techniques for solving First Order ODE and Second Order Linear ODE,
including Cau
Math 257/316 Assignment 10 Solutions
1. Find the eigenvalues and eigenfunctions of the following Sturm-Liouville problem on
[0, 1]
X (x) + X(x) = 0, X (0) = 0, X(1) = 0,
and use them to solve the foll
Math 257/316 Assignment 6 Solutions
1. For the function
f (x) =
1
1
2x 0 x < 1/2
1/2 x 1
dened on [0, 1], sketch (several periods of) its even and odd 2-periodic extensions,
and for each x 2 [0, 1] de
Math 257/316 Assignment 10
Due: Fri. Apr. 10
1. Find the eigenvalues and eigenfunctions of the following Sturm-Liouville problem on
[0, 1]
X (x) + X(x) = 0, X (0) = 0, X(1) = 0,
and use them to solve
Math 257/316 Assignment 8
Due Mon. Mar. 23 in class
1. Consider the wave equation utt = c2 uxx , < x < , with initial position
x + 1 if 1 < x 0
1 x if 0 < x < 1
u(x, 0) = f (x) =
and with initial ve
Math 257/316 Assignment 7
Due Monday Mar. 9 in class
1. The concentration u(x, t) of a reactive chemical diusing in one dimension satises
0 < x < 2, t > 0
ut = uxx u,
u(0, t) = 1, u(2, t) = 1
.
u(x,
Math 257/316 Assignment 7 Solutions
1. The concentration u(x, t) of a reactive chemical diusing in one dimension satises
0 < x < 2, t > 0
ut = uxx u,
u(0, t) = 1, u(2, t) = 1
.
u(x, 0) = 0
where the
Math 257/316 Assignment 6
Due Monday Mar. 2 in class
1. For the function
f (x) =
1 2x 0 x < 1/2
1
1/2 x 1
dened on [0, 1], sketch (several periods of) its even and odd 2-periodic extensions,
and for e
Math 257/316 Assignment 5 Solutions
1. For the triangle wave function
x
0x1
2x 1x2
f (x) =
dened on [0, 2]:
(a) compute its Fourier sine series
Doing some integration by parts,
bk =
2
2
1
2
f (x) sin(
Math 257/316 Assignment 4 Solutions
1. Consider the heat conduction problem:
u
2u
= 5 2,
t
x
with homogeneous boundary conditions
0 < x < 3, t > 0,
u(0, t) = u(3, t) = 0.
Find the solution for each of
Math 257/316 Assignment 3
Due Friday Jan 30 in class
1. The steady-state temperature distribution y(x) along a wire 0 x 1, cooled by its
surroundings, and held xed at zero temperature at its left endp
Math 257/316 Assignment 2
Due Friday Jan 23 in class
Problem 1. Find the rst six non-zero terms in the power series y = an (x x0 )n of
n=0
the general solution of the following second-order, linear, h
Math 257/316 Assignment 2 Solutions
Problem 1. Find the rst six non-zero terms in the power series y = an (x x0 )n of
n=0
the general solution of the following second-order, linear, homogeneous ODEs,
Math 257/316 Assignment 5
Due Monday Feb. 23 in class
1. For the triangle wave function
f (x) =
x
0x1
2x 1x2
dened on [0, 2], compute its
(a) compute its Fourier sine series
(b) computes its Fourier c