M257/316 SolutionsAssignment 5
UBC M257/316 Resources (c) 2014 by Philip D. Loewen
1.
PART I: At the initial time t = 0, we want f = u(0), i.e.,
10
8 = a1 (0)u1 + a2 (0)u2 + a3 (0)u3 .
9
As shown in Assignment 4, Question 3, this implies
a1 (0) = 3,
a2 (
Math 257/316 (sec. 103) Assignment 7
Due: Wednesday, November 10
1. There is a gas leak at one end of a long corridor 0 < x < 1. The concentration of gas satises
ut = uxx ,
0<x<1,
with boundary conditions and initial condition
ux (0, t) = 0 ,
ux (1, t) =
Math 257/316 Assignment 5
Due: Friday, October 22
1.
(a) Compute the Fourier Sine Series of f (x) = x3 x on [0, 1].
(b) Evaluate the series found in (a) at x = 1/2 to obtain
1
1
1
3
= 1 3 + 3 3 + .
32
3
5
7
(c) Drop all but the rst two terms in the above
Math 257/316, Midterm 1, Section 101/102
8 October 2010
Last Name:
First Name:
Student Number:
Instructions. The exam lasts 55 minutes. Calculators are not allowed. A formula sheet is attached.
1
1. Consider the ODE
2x2 y + (x + )y (x + 1)y = 0,
in which
Math 257/316 Assignment 6
Due: Friday, October 29
1.
(a) Find the solution (in series form) of the following heat conduction problem with homogeneous Dirichlet boundary conditions:
ut = 2uxx , (0 < x < 2, t > 0)
u(0, t) = u(2, t) = 0 ,
0
0x1
u(x, 0) =
.
1
Name 30 0
UBC Student Number 3' I 2
Signature
The University of British Columbia
Midterm Examination 19 June 2014
Mathematics 257/316
Dieiential Equations II
Closed book examination Time: 110 minutes
Special Instructions:
No calculators
M257/316 SolutionsAssignment 2
UBC M257/316 Resources (c) 2014 by Philip D. Loewen
1. Since the initial data are given at the point x = 0, we use that point as the expansion
centre for the desired series solution:
y=
a k xk .
k=0
This series satises the d
M257/316 SolutionsAssignment 1
UBC M257/316 Resources (c) 2014 by Philip D. Loewen
1
for each constant z satisfying |z| < 1, and that the
1z
k=0
series diverges whenever z obeys |z| 1. Careful accounting for the lowest exponent
involved is often needed to
M257/316 SolutionsAssignment 3
UBC M257/316 Resources (c) 2014 by Philip D. Loewen
1. The function y satises the given dierential equation if and only if
ty ( t) + (4t 1)y ( t) + 20t3/2 y( t) = 0,
t > 0.
Dene u(t) = y( t). Then the Chain Rule gives
1
u
M257/316 SolutionsAssignment 4
UBC M257/316 Resources (c) 2014 by Philip D. Loewen
1. (a) Its hard work to nd eigenvectors, but its pretty easy to test a specic vector
to see if it satises the dening property. By multiplying
26 11
1
1
7
(1)
11
2 = 14
ASMSW 4 W5
04va Q) W 03 Maria. [945%
6194 1e W8. HL i5 jVavvij
M '1. ML Wt M 62H)=Cek. We 524/4.
5mm, M Q(9)=3000 M 0(5)::0000. Mir-M
Cemtsooo (I)
Ce:+ '3 {3)
Dwight; {25 57(1), m, mm, e3*=5-=>3+=m:=>t=%fhs
Subs/HM-j bah: (I), M: 3%
9/3 A?
C =
Math 257/316 Assignment 4
Due: Friday, October 15
1. Determine the minimal (fundamental) period of
a)
sec 3x,
b)
sin2 x,
c)
sin
x
x
+ sin .
4
7
2. Sketch the following functions over two periods, and nd their Fourier series:
a) f (x) =
x + 1,
1 x < 0,
1 x
Math 257/316 Assignment 8
Due: Wednesday, November 24
1. Solve Laplaces equation, uxx +uyy = 0, in the unit square, cfw_(x, y) | 0 < x < 1, 0 < y < 1,
with the following Neumann boundary conditions:
ux (0, y)
ux (1, y)
uy (x, 0)
uy (x, 1)
=
=
=
=
0,
cos(2
Math 257/316 Assignment 9
Due: Friday, Dec 3
1.
a) Consider the following Sturm-Liouville eigenvalue problem
X + 2 X = 0,
X(0) = 0,
X (1) = X(1).
Find the eigenfunctions and the transcendental equation of which the eigenvalues n
are the roots. Show graphi