Homework for Math 46
Homework assignments will generally be due at the class time on
Wednesday of the week following the week when the homework was assigned, unless
announced otherwise. Late homework will not be accepted unless it
is excused in advance for serious reasons. Unexcused late and missing papers
count zero.
Lectures 
Sections in Text 
Homework
assignment 
Day 1, Monday March 29 
Read Sections 1.1.2 and 1.1.3 
Exercise 1 on page 7; Attention there is a typo, it should be P instead of T in the problem formulation. Exercise 4 page 8 (In this problem choose π_{1 }from the computation on page 6 and choose π_{2 }that does not involve the distance r.) Exercise 3 page 17 (First take the speed v to the left to get that some expression equals zero.) Exercise 6 page 17; due Wednesday April 7 in
written form Sketches of solution can be found here 
Day 2, Wednesday March 31 
Read Section 1.1.3 and Section 1.2 
Exercise 3 on page 30 and Exercise 12 on page 33. (Hint: in exercise 12 use ℓ rather than L for the initial condition, so that there are no mixtures between the initial conditions and dimensions. When rescaling the problem choose x_{c}=ℓ and do not attempt to solve the equation.) due Wednesday April 7 in
written form Sketches of solution can be found here 
Day 3, Thursday April 1 xhour instead of the lecture on Friday April 16 
Read Section 1.2 
No additional homework 
Day 4, Friday April 2 
Read Section 1.3.1 
Exercise 1.c on page 40 (will be counted as one exercise when grading). Note that the book formulation has a typo, you should change y to u in this problem. You do not have to compute the integral of t^{2}exp(t^{2}/2) Exercise 1.d and 1.e on page 40 (together are to be counted as one exercise when grading). In 1.e if the integrals are hard do not compute them and leave them as the part of the answer. Exercise 1.f and 1.j on page 40 (together are to be counted as one exercise when grading). In these problems you do not have to compute the integrals of exp(s^{2}) and of 1/(s^{2}+c) for constants c of various signs. Exercise 1.n on page 40 will be counted as one exercise when
grading. due Wednesday April 7 in
written form Sketches of solution can be found here 
Day 5, Monday April 5 
Read Section 4.1.1 
Exercise 1, page 214. The following part will be graded as one problem: Show that the functions f_{m}(x)=cos(mπx/ ℓ) form an orthogonal family in L^{2}[0, ℓ]. Find the explicit formula for the Fourier coefficients of f with respect to this family. Hint: use the integral formula ∫cos(ax)cos(bx)dx=(sin(ab)x)/(2(ab))+ (sin(a+b)x)/(2(a+b)) when a^{2}≠ b^{2} Exercise 1, page 214. The following part will be graded as one problem. Find the Fourier cosine series of f(x)=1x on [0,1]. Exercise 3, page 214 due Wednesday April 14 in
written form Sketches of solution can be found here 
Day 6, Wednesday April 7 
Read Section 4.1.1 and Section 4.1.2 
Show that L^{2 }is a vector space. (Hint you may want to use the Cauchy Schwarz inequality.) Simplified version of Exercise 5, page 215. Find P_{0}(x), P_{1}(x), P_{2}(x), P_{3}(x). Approximate e^{x }as c_{0}*P_{0}(x)+c_{1}*P_{1}(x)+c_{2}*P_{2}(x). Do not compute errors and do not sketch anything. Exercise 1, page 219. Do not sketch anything. Hint: your life will be much easier if you will use various facts about integration of odd and even functions. due Wednesday April 14 in
written form Sketches of solution can be found here 
Day 7, Thursday April 8 xhour 
Read Section 4.1.2 
No homework 
Day 8, Friday April 9 End of the second week of schedule adjustment 
Read Section 4.2 
Exercise 3, page 225 Exercise 4, page 225 Exercise 7, page 225. Use energy argument to show that the eigenvalues have to be nonnegative. Do not find eigenvalues or eigenfunctions in Exercise 7. due Wednesday April 14 in
written form Sketches of solution can be found here 
Day 9, Monday April 12 Final day for electing use of
the NonRecording option 
Section 4.3.1 
Exercise 1, page 243 Exercise 2 parts a and b, page 243 (these two parts combined will be counted as one exercise when grading) Exercise 3, page 244. The answer should be given in terms of whether the vector b does belong or does not belong to some explicitly given vector subspace. due Wednesday April 21 in written
form Sketches of solution can be found here 
Day 10, Wednesday April 14 
Section 4.3.3 
Exercise 4.b page 244. Hint rewrite K(u(x)) as an integral from 0 to x of some expression plus the integral from x to 1 of some expression. Obtain the differential equation by differentiating the equation K(u(x))=λu(x) twice using the Leibnitz formula from Exercise 1. Get the conditions on u(0) and u’(1). Exercise 13 part a, page 245. Hint: this is a integral equation with a separable kernel. Exercise 16, page 245. When the book says two terms in its power
series expansion, it means the due Wednesday April 21 in
written form Sketches of solution can be found here 
Friday April 16, No class, instead we had an xhour on April 1 


Day 11, Monday April 19 
Section 4.3.4 and start 4.3.2 
Exercise 13 part b page 245 Exercise 15, page 245 Exercise 18, page 246 Due Wednesday April 28 in written form Problems of this type may appear on the
first Midterm. Volterra equations will not be
included into the Midterm material. Sketches of solution can be found here 
Day 12, Wednesday April 21 
Section 4.3.2 
Exercise 6, page 244 Exercise 8, page 244 Exercise 10, page 245 Due Wednesday April 28 in written form Sketches of solution can be found here 
Thursday April 22 First Midterm Exam 68 PM in Carpenter 013 


Day 13, Friday April 23 
Section 4.4.1 
Exercise 3, page 257 Exercise 4, page 257 In these problems you may want first to multiply the equations by a function to make them into an SturmLiouville problem. Exercise 5, page 257. In these problems write the expression for Greens function g(x,ζ) AND write the solution of the differential equation as an integral involving the Green’s function g(x,ζ) Due Wednesday April 28 in written form Sketches of solution can be found here 
Day 14, Monday April 26 
Section 4.4.1 and 4.4.3 
Exercise 2, page 257 If there is no Green’s function find for which functions f(x) the equation can be solved. Find the appropriate function G(x,ζ) described in Theorem 4.23 and give an answer in terms of this function. Exercise 7, page 258 Due Wednesday May 5 in written form Sketches of solution can be found here 
Day 15, Wednesday April 28 
Section 4.4.3 and Section 4.5.1 
Exercise 3 on page 267 Due Wednesday May 5 in written form Sketches of solution can be found here 
Day 16, Thursday April 29 xho 
Section 4.5.1 

Day 17, Friday April 30 
Section 4.5.2 and Section 4.5.3 
Exercise 1 page 267, Exercise 5, page 268 Exercise 8, page 268 Exercise 9, page 268 Exercise 11, page 268 In these problems unless the interval is explicitly stated you can assume that it is (a, a) for a finite positive a. Due Wednesday May 5 in written form Sketches of solution can be found here 
Day 18, Monday May 3 
Section 4.5.3 and Section 6.1.1 
Exercise 4, page 267 Exercise 6, page 268 Exercise 3, page 345 Exercise 2 part a, page 345 Exercise 2 part b, page 345 (these two parts of exercise 2 will be counted as different problems when graded) Hint: in exercise 2 it is useful to use the following trick fix one of the variables say t=τ and denote u(x, τ) by a function g _{τ}(x). This would allow you to convert a PDE to an ODE. Be careful the constants you get when solving this ODE are in fact functions depending on the fixed t=τ. Due Wednesday May 12 in written form Sketches of solution can be found here 
Day 19, Wednesday May 5 
Section 6.1.1 and Section 6.2.1 
Exercise 2, part e on page 345 Exercise 5, page 346 Exercise 7, page 346 Due Wednesday May 12 in written form Sketches of solution can be found here 
Day 20, Friday May 7 Final day
for dropping a fourth course without a grade notation of "W" 
Section 6.2.2 and Section 6.2.5 
Exercise 2 on page 365 Exercise 3 on page 365 (note that the domain Ω in this problem is a domain in R^{n}) Due Wednesday May 12 in written form Sketches of solution can be found here 
Day 21, Monday May 10 
Section 6.2.2 and Section 6.2.3 
Page 352: Learn and understand the proofs of statements (iii), (iv), (v), and the proof of how to get the vector form of the divergence theorem from the other one Exercise 10 on page 366 (do not do the physical interpretation) Exercise 13 on page 367 Due Wednesday May 19 in written form Problems and proofs of this type may appear
on the second Midterm. Sketches of solution can be found here 
Day 22, Wednesday May 12 
Sections 6.2.4, 6.3.1, and 6.3.2 
Exercise 4, page 372 Exercise 5, page 372 Exercise 6, page 372 Due Wednesday May 19 in written form Sketches of solution can be found here 
Thursday May 13 Second Midterm Exam 68 PM in Carpenter 013 


Friday May 14 No class. We shall have an xhour on Thursday May 20 instead 

^{ } 
Day 23, Monday May 17 Note that Tuesday May 18 is the final day to withdraw from
a course 
Section 6.3.2 and 6.4.1 
In the following three problems Ω=[0,π]× [0,π] Exercise 1: Solve the equation –Δu=1 for x in Ω; and u=0 for x in boundary of Ω; Exercise 2: Solve the equation –Δu=3sin(4x)sin(3y) for x in Ω; and u=0 for x in boundary of Ω; Exercise 3: Find the eigen values and the eigen functions corresponding to the homogeneous Neumann problem: –Δu=λu for x in Ω; and du/dn=0 for x in boundary of Ω;
Due Wednesday May 26 in written form Sketches
of solution can be found here 
Day 24, Wednesday May 19 
Section 6.4.1 and Section 6.4.2 
Exercise 1 part a, page 381 Exercise 1 part b, page 382 These two parts will be counted as separate exercises when the homework will be graded. Due Wednesday May 26 in written form Sketches of solution can be found here 
Day 25, Thursday May 20 xhour instead of the class on Friday May 14 
Section 6.4.2 
No homework 
Day 26, Friday May 21 
Section 6.5.2 
Exercise 5, page 396 Exercise 6, page 396 You may assume that the constant a in problem 5 is positive, since otherwise the integral in the definition of the Fourier transform will not converge. Due Wednesday May 26 in written form Sketches of solution can be found here 
Day 27, Monday May 24 
Section 6.5.2 
Exercise 7, page 396 Hint: in this exercise you may want to use the completion of the square technique used in deriving the formula for the solution of the quadratic equation. You also may want to use Exercise 5 part b on page 396. Exercise 11, page 397 Exercise 13, page 397 Due Wednesday June
2 in written form Sketches of solution can be found here 
Day 28, Wednesday May 26 
Section 6.5.2 
Exercise 8, page 397 (In this exercise you do not have to compute the inverse Fourier transform.) Exercise 9, part b page 397 Exercise 9, part c page 397 The two parts of exercise 9 will be counted as separate exercises when graded. To find the inverse Fourier transform in Exercise 9.c one has to introduce a convolution (u*v)(x) of two functions u,v:R^{n}→R, via the formula (u*v)(x)=∫u(y)v(xy)dy, where the integral is taken over R^{n}. Then one shows that F(u*v)(ζ)=(F(u)(ζ))×(F(v)(ζ)), similarly to Theorem 6.32 Due Wednesday June 2 in written form Sketches of solution can be found here 
Day 29, Friday May 28 
Section 6.5.2 and sections 6.7.1, 6.7.2 
Exercise1: Part A. Find the second distributional derivative of the distributional Fourier transform of δ(xx_{0}). Part B. Find the distributional Fourier transform of the second distributional derivative of δ(xx_{0}). Exercise 2: Use the y^{(4)}(x)4y(x)=0 the
initial conditions are: y(0)=1, y’(0)=0, y’’(0)=2,
y’’’(0)=0. You may want to use the method of partial
fractions and table on page 384 to help you find the inverse Exercise 1, page 415 Exercise 2, page 415 Due Wednesday June 2 in written form Sketches of solution can be found here 
Monday May 31 Memorial Day, No Class 


Day 30, Wednesday June 2 Last day of classes 
Section 6.7.2 and section 6.7.3 
No homework 
Friday June 4 Final Exam 36 PM in Kemeny 105 

