Solutions to Homework # 4 Math 381, Rice University, Fall 2003 Problem 1: We start by making the Euler substitution t = ln x, or x = et . Then we compute that dy = xy , dt and d2 y = x2 y + xy . dt2
So we can rewrite our original equation as dy d2 y + ( -
Math 381: Final Exam
December 11, 2003 2pm - 5pm
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Read the directions to each problem carefully. Show your work and answer in complete sentences when appropriate. This exam has 5 questions on 11 pages including this cover, blank pages for work and the ta
Solutions to the Final Examination Math 381: Introduction to Partial Differential Equations Rice University, Fall 2003
Problem 1: a) This is a fourth order equation. b) Many correct responses are possible. An example would be z z + 12 = 3z. x y c) Again,
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Math 381: Second Exam Due: Monday, November 10, 2003 at the beginning of class. This exam is self-timed. You have a maximum of 3 hours. You may consult your textbook or your class notes, but you may not discuss the content of this exam with anyone but y
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Math 381: Second Exam
November 5-10, 2003 Solutions Problem 1: We rewrite our problem in polar coordinates because these are adapted to the disk D. Then our boundary condition is g(1, ) = (cos )2 . Following the developments in class, we know that the s
Math 381: First Exam
September 26, 2003 Solutions Problem 1: The equations are second order, fourteenth order and first order, respectively.
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Problem 2: We compute the derivatives 2y = f (x + at) - g (x - at), x2 2y = a2 f (x + at) - a2 g (x - at). t2 So
Solutions to Homework # 1 Math 381, Rice University, Fall 2003 Hildebrand, Ch. 8, # 1: Part (a). We compute z = f (x + y) + (x - y)f (x + y) x z = -f (x + y) + (x - y)f (x + y). y Subtracting, we eliminate f . . . z z - = 2f (x + y). x y Substitute in fro
Solutions to Homework # 2 Math 381, Rice University, Fall 2003 Problem1: The function is odd, so all of the ak 's are zero, including a0 . We then compute the bk 's to be bk = = 2
sin(kx)dx =
0
-2 -2 (cos(k) - cos(0) = (-1)k - 1 k k
0,
4 k ,
k even . k o
Solutions to Homework # 5 Math 381, Rice University, Fall 2003 Problems 1-4: Since these are polynomials, it is not to hard to compute the answers. First, note that a Legendre polynomial is orthogonal to any polynomial of lower degree, so most of the coef
Solutions to Homework # 7 Math 381, Rice University, Fall 2003 Problem 1: We work each by hand using integration by parts. a) We integrate by parts twice to find that L(eat cos(kt) =
0
e-st eat cos(kt) dt e-(s-a)t cos(kt) dt
=
0
-e-(s-a)t cos(kt) ke-(s-
Solutions to Homework # 6 Math 381, Rice University, Fall 2003 Problem 1: The problem is about Laplace's equation in a sphere, so we shall use spherical coordinates (r, , ). First, we translate our boundary condition. Note that in spherical coordinates g(
Math 381: First Exam
September 26, 2003
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Read the directions to each problem carefully. Show your work when necessary. This exam has 8 questions on 5 pages and lasts 50 minutes. Name: Pledge:
Problem 1:(6 points) Identify the order of the following parti