Final Exam
MATH 150: Introduction to Ordinary Differential Equations
27th May 2010
Answer ALL questions
Full mark: 100
Time: 16:30-18:30
This is a closed book exam. You may write on the front and back of the exam papers.
Student Name:
Student ID Number:
L
HKUST
MATH150 Introduction to Dierential Equations
Final Examination (Version A)
Name:
26th May 2007
Student I.D.:
Tutorial Section:
08:3010:30
Directions:
Write your name, ID number, and tutorial section in the space provided above.
DO NOT open the exa
HKUST
MATH150 Introduction to Dierential Equations
Final Examination (Version A)
Name:
Student I.D.:
25th May 2006
8:30am10:30am
Tutorial Section:
Directions:
Write your name, ID number, and tutorial section in the space provided above.
DO NOT open the
HKUST
MATH150 Introduction to Dierential Equations
15th Dec 2004 Final Examination Solution (Version White)
Part I: Each correct answer in the answer box for the following 8 multiple choice questions is
worth 4 point. DO NOT guess wildly! If you do not ha
Math 150, L3, L4, Final Exam, Spring 2002
Date:
Time:
24 May 2002
4:30p.m.|6:30p.m.
Venue:
G017
Name
Student Number
Tutorial Section
Score
1. A tank with a capacity of 800 gal originally contains 300 gal of water with 200
lb of salt in solution. Water con
Math 150
Introduction to Ordinary Dierential Equations
Final Examination
December 16, 4:30pm-7:30pm
Section A: Answer all questions from this section.
1. Consider a pond that initially contains 10 gal of fresh water. Water
containing an undesirable chemic
Math 150
Introduction to Ordinary Dierential Equations
Final Examination
Fall 2002
December 16, 4:30pm-7:30pm
Section A: Answer all questions from this section.
1. Consider a pond that initially contains 10 gal of fresh water. Water
containing an undesira
Math 150, Final Exam, Solutions, Spring 2001
1. Solve the following Euler's equation (8pts)
t2 y + 5ty + 13y = 0
00
Solution.
t > 0:
0
Let
z = ln t:
Then
dy = dy dz = 1 dy d2y = 1 d2y ; dy :
dt dz dt t dz dt2 t2 dz2 dz
The equation becomes
d2y ; dy + 5 dy
Numerical answers for note 6
(1)
Forward difference: f '(0.68) 11.658
Backward difference: f '(0.68) 40.930
Forward difference: f '(0.69) 1.313
Backward difference: f '(0.69) 11.658
(2)
Equation (1): Error bound =
K
0.0033333K
300
K
0.0016667 K
600
Equa
Numerical answers for note 5
(1)
(2)
Forward difference: f '(0.68) 11.658
Backward difference: f '(0.68) 40.930
Forward difference: f '(0.69) 1.313
Backward difference: f '(0.69) 11.658
Equation (1): Error bound =
K
0.0033333K
300
K
0.0016667 K
600
Equa
Numerical answers for note 3
(1)
(2)
P2 ( x) = 0
( x 1)( x 2)
( x 0)( x 2)
( x 0)( x 1)
+ 1
+ 8
(0 1)(0 2)
(1 0)(1 2)
(2 0)(2 1)
( x (1)( x 0)( x 1)( x 2)
(2 (1)(2 0)(2 1)(2 2)
( x (2)( x (1)( x 1)( x 2)
+ 0 .7
(1 (2)(1 0)(1 1)(1 2)
( x (2)( x (1)( x 1)
Numerical answers for note 1
(1)
(2)
p4 = 0.85085
p10 = 0.64160
(3)
(a) 2
(b) 2
(c) 1
(4)
We define g ( x) =
x + cos x
and p4 = 0.73912 . (You may get different p4 's
2
for other definitions of g(x).)
(5)
Assume we dont know p = 3 ,
know what ps are.)
= 1
B: MATLAB Introduction
The name MATLAB stands for matrix
laboratory.
Features:
Math and computation
Algorithm development
Data acquisition
Modeling, simulation, and prototyping
Data analysis, exploration, and visualization
Scientific and engineering graph
MATH230
Tutorial note 11
1. Introduction
Given a matrix A and a vector b, both known, we can use Gaussian Elimination for
finding the vector x such that
Ax = b .
However if the A is very large and sparse (the number of nonzero elements is a small
fraction
MATH230
Tutorial note 10
1. Introduction
Solving sets of linear of equations is the most frequently used numerical procedure
when real-world situations are modeled. Linear equations are the basis for
mathematical models foe economics, weather predictions,
MATH230
Tutorial note 9
1. Initial value problem (IVP)
Consider the ordinary differential equation (ODE)
dy
= f ( t , y (t ) )
dt
y (t0 ) = y0
N +1
where f is a function from
into N for some N > 0 (if N = 1, then we have a
scalar equation; otherwise a
MATH230
Tutorial note 8
1. Composite Numerical Integration
b
Suppose f C 2 [a, b] . In order to approximate
f ( x)dx ,
we can subdivide the
a
interval [a,b] into n subintervals and apply Trapezoidal rule on each subinterval. That
is, we let h =
ba
and xk
MATH230
Tutorial note 7
1. Numerical Integration
b
f ( x)dx ,
Given a function f which is continuous on [a,b]. If we are asked to evaluate
a
we can try to find an antiderivative of f, F ( x) , and then apply the formula
b
f ( x)dx = F (b) F (a) .
a
2
Un
MATH230
Tutorial note 6
1. Numerical differentiation
Give a function which is differentiable, one can always differentiate it ready. However,
equations with derivatives, that is differential equations, are rarely solvable. Because
of the importance of the
MATH230
Tutorial note 5
1. Data fitting
An important area in approximation is the problem of fitting a curve to experimental
data. Since the data is experimental, we must assume that it is polluted with some
degree of error, most common measurement error,
MATH230 Tutorial note 4
1. Divided differences
The Lagrange form of the interpolating polynomial gives us a very tidy construction,
but it does not lead itself well to actual computation. One of the reasons is that
whenever we decide to add a point to the
MATH230
Tutorial note 3
1. Interpolation
One of the oldest problems in mathematics is the problem of construction an
approximation to a given function f from among simple functions, typically (but not
always) polynomial. A slight variation of this problem
MATH230 Tutorial note 2
1. Newtons Method
Newtons method is the classic algorithm for finding roots of functions. It appears to
have been first used by Newton in 1669, although the ideas were known to others
beforehand.
Suppose f '( x) exists on [a,b] and