CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 12 Solutions
Posted Friday 26 November 2010. Due Friday 3 December 2010, 5pm.
This problem set counts for 100 points, plus a 20 point bonus.
1. [50 points: 15 points for (a); 10 points for (b); 5 points for (c);
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 11 Solutions
Posted Thursday 18 November 2010. Due Tuesday 23 November 2010, 5pm.
This problem set counts for 75 points.
Late problem sets are due by 5pm on Wednesday 24 November 2010.
1. [30 points: 7 points ea
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 10 Solutions
Posted Wednesday 3 November 2010. Due Wednesday 10 November 2010, 5pm.
1. [50 points: 8 points each for (a), (b), (d), (e); 4 points for (c); 14 points for (f)]
This problem and the next study the h
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 9 Solutions
Posted Thursday 28 October 2010. Due Wednesday 3 November 2010, 5pm.
1. [40 points: 12 points each for (a) and (b); 6 points for (c); 10 points for (d)]
1, x [0, 1/3];
0, x (1/3, 2/3);
(a) Consider
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 8 Solutions
Posted Thursday 21 October 2010. Due Wednesday 27 October 2010, 5pm.
1. [50 points: 18 points for (a); 12 points for (b); 10 points each for (c) and (d)]
Consider the following three matrices:
(i) A
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 7 Solutions
Posted Wednesday 6 October 2010. Due Wednesday 13 October 2010, 5pm.
General advice: You may compute any integrals you encounter using symbolic mathematics tools such as
WolframAlpha, Mathematica, or
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 6 Solutions
Posted Wednesday 29 September 2010. Due Wednesday 6 October 2010, 5pm.
General advice: You may compute any integrals you encounter using symbolic mathematics tools such as
WolframAlpha, Mathematica,
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 5 Solutions
Posted Thursday 23 September 2010. Due Wednesday 29 September 2010, 5pm.
All of the problems on this set use the inner product
1
(u, v ) =
0
u(x)v (x) dx.
1. [30 points: 6 points each for (a) and (b)
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 4 Solutions
Posted Wednesday 15 September 2010. Due Wednesday 22 September 2010, 5pm.
1. [20 points]
The equation x1 + x2 + x3 = 0 denes a plane in
(a) Find two linearly independent vectors in
R3 that passes thr
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 3 Solutions
Posted Wednesday 8 September 2010. Corrected 14 September. Due Wednesday 15 September 2010, 5pm.
1. [20 points: 5 points per part]
Determine whether each of the following functions (, ) determines an
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 2 Solutions
Posted Wednesday 1 September 2010. (Corrected 6 Sept. 2010.) Due Wednesday 8 September 2010, 5pm.
1. [27 points: 4.5 points per problem]
Consider the following sets of functions. Demonstrate whether
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 1 Solutions
Posted Wednesday 25 August 2010. Due Wednesday 1 September 2010, 5pm.
1. [18 points: 3 points per part]
For each of the following equations, specify whether each is (a) an ODE or a PDE; (b) determine
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 8
Posted Wednesday 27 February 2008. Due Friday 7 March 2008 in class.
1. [50 points]
Use the nite element method to solve the dierential equation
d
du
(1 + x2 )
dx
dx
= 2x,
0<x<1
subject to the inhomogeneous bo
CAAM 336
DIFFERENTIAL EQUATIONS IN SCIENCE AND ENGINEERING
Examination 1
SOLUTIONS
1. [20 points]
Vector spaces
(a) Determine if the following sets are vector spaces or not- justify your answer. (Dene
addition and scalar multiplication in the obvious way.
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 13
Posted Friday 11 April, 2008. Due Friday 18 April, 2006 in class.
1. [50 points]
We wish to approximate the solution to the homogenous wave equation
2u
2u
c2 2 = 0 ,
t2
x
0 x 1, t 0
with homogeneous Dirichle
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 12
Due Wednesday 9 April 2008 in class.
1. [60 points]
We wish to approximate the solution to the heat equation
u 2 u
2 = 100 tx,
t
x
0 x 1, t 0
with homogeneous Dirichlet boundary conditions
u(0, t) = u(1, t)
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 12
Posted Friday 26 November 2010. Due Friday 3 December 2010, 5pm.
This problem set counts for 100 points, plus a 20 point bonus.
1. [50 points]
On Problem Set 10, you solved the heat equation on a two-dimensio
CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 11
Posted Wednesday 26 March 2008. Due Wednesday 2 April 2008 in class. 1. [50 points] Consider the wave equation 2u 2u = c2 2 t2 x for 0 x 1 and t 0 subject to the mixed boundary conditions u(0, t) = 0, for all
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 11
Posted Thursday 18 November 2010. Due Tuesday 23 November 2010, 5pm.
This problem set counts for 75 points.
Late problem sets are due by 5pm on Wednesday 24 November 2010.
1. [30 points: 7 points each for (a)
CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 10
Posted Wednesday 19 March 2008. Due Wednesday 26 March 2008 in class. 1. [40 points] For each of the following examples, (1) compute the matrix exponential etA by hand, and (2) explain the behavior of dx/dt =
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 10
Posted Wednesday 3 November 2010. Due Wednesday 10 November 2010, 5pm.
1. [50 points: 8 points each for (a), (b), (d), (e); 4 points for (c); 14 points for (f)]
This problem and the next study the heat equati
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 9
Posted Friday 14 March 2008. Due Friday 21 March 2008 in class.
1. [70 points]
Use the Fourier series method to solve the following initial boundary value problem
u 2 u
2
t
x
u
(0, t)
x
u
(1, t)
x
u(x, 0)
= 4x
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 9
Posted Thursday 28 October 2010. Due Wednesday 3 November 2010, 5pm.
1. [40 points]
1, x [0, 1/3];
0, x (1/3, 2/3);
(a) Consider the function u0 (x) =
1, x [2/3, 1].
2
Recall that the eigenvalues of the opera
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 8
Posted Thursday 21 October 2010. Due Wednesday 27 October 2010, 5pm.
1. [50 points]
Consider the following three matrices:
(i) A =
0
1
1
0
(ii) A =
50
49
49
50
(iii) A =
0
1
1
.
0
(a) For each of the matrices
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 7
Posted Wednesday 6 October 2010. Due Wednesday 13 October 2010, 5pm.
General advice: You may compute any integrals you encounter using symbolic mathematics tools such as
WolframAlpha, Mathematica, or the Symbo
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 6
Posted Wednesday 29 September 2010. Due Wednesday 6 October 2010, 5pm.
General advice: You may compute any integrals you encounter using symbolic mathematics tools such as
WolframAlpha, Mathematica, or the Sym
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 5
Posted Thursday 23 September 2010. Corrected 28 Sept 2010. Due Wednesday 29 September 2010, 5pm.
All of the problems on this set use the inner product
1
(u, v ) =
u(x)v (x) dx.
0
1. [30 points]
2
Consider the
CAAM 336 DIFFERENTIAL EQUATIONS
Problem Set 4
Posted Wednesday 15 September 2010. Due Wednesday 22 September 2010, 5pm.
1. [20 points]
The equation x1 + x2 + x3 = 0 denes a plane in
(a) Find two linearly independent vectors in
R3 that passes through the o