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. (15 points) Find the solution of the non-homogeneous
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 space and one in time. The spatial ODE and boundary con
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 in the form of a power series based at x0 = 0 (just
nd t
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. (15 points) Find the solution of the non-homogeneous init
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 deliberately explore the dierent paths to arrive at th
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 computer.
We use the denition of the derivative and Taylor ser
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 the PDE to solving two ODEs, one in time and one in space
Math 257/316 Section 202 Midterm 1 February 6
2 questions; 50 11ii11utos; max r:- 30 points
IL Consider this second order, linear, homogeneous ODE:
2&3 -- 1)y + 11+?) : 0.
(FL) Find the genera: solution in the form of a power series based at 1130 = U (g
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 classication of these equations we go through the modeling proc
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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
of separation of variables for problems with radial or
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 initial velocity ut (x, 0) = 0.
:
;
0
otherwise
Sketch t
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 the series and how it relates to the coecient of the h
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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 about which Taylor series solutions are obtained and sing
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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 approximations are appropriate, regular singular points for which F
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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 Cauchy-Euler/Equidimensional Equations
Key Concepts: First
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 following heat equation with mixed BCs and a source term:
u
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] determine the value to which its Fourier sine and cosine
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 the following heat equation with mixed BCs and a source
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 velocity ut (x, 0) = 0.
0
otherwise
Sketch the shape of t
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, 0) = 0
where the loss term represents a reaction which
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 loss term represents a reaction which consumes the chem
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 each x [0, 1] determine the value to which its Fourier s
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(kx/2)dx =
2
(2 x) sin(kx/2)dx
x sin(kx/2)dx +
0
1
0
1
2
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 the initial conditions (using formulas from class/note
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 endpoint, solves:
d
p(x)y y = 0,
y(0) = 0,
dx
where p(x) 0
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, homogeneous ODEs, centred at
the indicated point x0 :
a)
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, centred at
the indicated point x0 :
a)
y + x2 y + y = 0
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 cosine series
(c) by evaluating f (1), use each of your