The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
Examination Review
16/10/02
Chapter13
Mock Questions
Chapter One
Analysis:
I guess that one of the questions may
be in this section
Q1.
a)Linear First Order Equations
b)Bernoulli Equations
First Chapter Basic Concept
y + p(t)y = g(t)
How to solve
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
MATH 152
Applied Linear Algebra and Dierential Equations
Home Work 4
Matrix, System of Linear Equations
0
1. (Inverse) Verify that A = 0
1
Then compute the inverses of A2
Solution By
Fall 2009
Suggested Solution
0 1
1 2
1 2 and B = 2 1
2 3
1
0
t
and A .
1
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
MATH 152
Applied Linear Algebra and Dierential Equations
Home Work 2
Secondorder Dierential Equations
Fall 2010
Suggested Solution
1. (Linear property for secondorder linear dierential equations)
(a) If y1 (t) and y2 (t) are two solutions of the linear
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
MATH 152
Applied Linear Algebra and Dierential Equations
Home Work 3
Laplace Transform
Fall 2009
Suggested Solution
1. (Initial value problem) Consider the initial value problem
3
1
y + y = e 2 t , y(0) = 1.
2
(a) From the denition of Laplace Transform sh
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
MATH 152
Applied Linear Algebra and Dierential Equations
Home Work 5
Vector Space, Eigenvalue Problem
1. (Null space) Find the null space of A if
Solution We simplify
3 6
A = 1 2
2 4
the matrix A
1 1 7
2 3 1
5 8 4
1 2 2
5R +R1
3 0 0 1
Ri Rj
0 0 0
Fall 20
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
MATH 2350 Applied Linear Algebra and Differential Equations
Week 12 Tutorial
1. Use the given substitutions to solve the ODEs:
(i) y 0 cos y + cos t sin y = cos t, u = sin y.
(ii) 2t2 yy 0 + y + 2ty 2 t2 y 3 = 0, v = ty.
2. Solve the following second orde
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
MATH 2350 Applied Linear Algebra and Differential Equations
Week 13 Tutorial
1. Solve the following systems of ODEs for general solutions:
4 1
1 2
1 2
0
0
0
(i) x =
x
(ii) x =
x
(iii) x =
x.
2 1
3 2
4 5
2. Find the equilibrium solutions of the system:
4 3
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
Qn. 1 (20 marks) Choose a correct option for each question. No justification is required. Each
correct answer is worth 4 marks (no deduction for wrong answers).
(i) Suppose that we have entered the following commands into GNU Octave:
> a=[1,2;3,4;5,6];b=[
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
Differential Equations (DEs):
Many physical laws are relations on rates, i.e. how fast
something is changing.
Newtons 2nd Law: F = ma
a = acceleration, rate of change in velocity.
Radioactive Decay: rate of decrease in N N .
Rate of change: described by
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
Definition of a Vector Space
A collection of vectors: V , scalars for scaling : R (or C);
A vector addition: +, A scalar multiplication:
for vector addition:
(i) (closure of vector addition) u + v is always in V .
(ii) (commutative law) u + v = v + u.
(i
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
MATH 152
Applied Linear Algebra and Dierential Equations
Firstorder Dierential Equations
Home Work 1
Fall 2010
Suggested Solution
1. (Linear vs nonlinear) Indicate whether the following equation is linear or nonlinear.
1.
dy
sin t = t2 y
dt
7. yy = t3 +
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
MATH 152
Applied Linear Algebra and Dierential Equations
Home Work 2
Secondorder Dierential Equations
Fall 2010
Due Date: 11 Oct. 2010
1. (Linear property for secondorder linear dierential equations)
(a) If y1 (t) and y2 (t) are two solutions of the lin
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
MATH 152
Applied Linear Algebra and Dierential Equations
Home Work 3
Laplace Transform
1. (Initial value problem) Consider the initial value problem
Fall 2010
Due Date: 18 Oct. 2010
3
1
y + y = e 2 t , y(0) = 1.
2
(a) From the denition of Laplace Transfor
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
1
Matrix
Additions and scalar multiplications are easy to understand. To understand the multiplications,
note that matrices and linear maps are in onetoone correspondence:
cfw_m nmatrices cfw_linear maps from Rn to Rm ,
under which matrix multiplicatio
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
CIVL 102
Fall 2007
Quiz 3
Traverse Adjustment by Constrained Optimization
Question 1
Consider the closed loop traverse shown in Figure 1 below. Angles are observed with a 3
theodolite, while each length observed is subject to a 5 mm standard error.
RO
L L
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
These notes serve as a summary and generalizations of what we have
learned in class. If you want, you can try to prove what has been stated
here.
1
Linear nthorder homogeneous dierential
equations
They are equations of the form:
y (n) + p1 y (n1) + + pn
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
Solutions of Math 152 Midterm Exam
1 (30 points)
(1) Find the general solution of equation
xdx + ydy = 0.
(2) Solve the initial value problem:
y t2 y = t2 ,
y(0) = 0.
Solution:
(1) Multiply the equation by 2, we get
d(x2 + y 2 ) = 0.
Therefore, we have
x2
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
This note is about the claim I made in class that the recipe given in the
textbook does not work when the coecients are not real numbers, but the
recipe I gave in class always works. Here is an example:
y iy = 2 sin t
(1)
Note that P = 2, Q = 0 and = 0 an
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
MATH 152
Applied Linear Algebra and Differential Equations
Syllabus  Fall 2010
Course Home Page
http:/www.math.ust.hk/~kchow/math152
Please check the course home page for news regarding the course.
Instructor
Dr. Keith K. C. Chow
Contact Details: Rm. 349
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
Math 152 Answer to Exercises of Chapter 1
Not absolutely guaranteed to be correct,
Exercise 1.2
Omitted = too complicated, we made a mistake
2
6 + cx
3. y = e2x x3 + ce2x
4. y = 2 + ce2t
x3
1
3u4 + 6u2 + 4c
4x3 + 3x2 6x + c
5. x = 3 + ce t
6. v =
7. y =
4
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
More PastPaper:
http:/ihome.ust.hk/~cs_gxx
Additional Problems For Math 152
True Statements
In 17, A is a 4 6 matrix, B and C are 6 6 matrices
1) The null space of A is a subspace of R6 ;
2) If Col A = R4 , then Ax = b has solution for all b in R4 ;
3) A
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
MATH 152
Applied Linear Algebra and Dierential Equations
Firstorder Dierential Equations
Home Work 1
Fall 2010
Due Date: 30 Sep. 2010
1. (Linear vs nonlinear) Indicate whether the following equation is linear or nonlinear.
1.
dy
sin t = t2 y
dt
7. yy =
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
MATH 152
Applied Linear Algebra and Dierential Equations
Home Work 5
Vector Space, Eigenvalue Problem
Due Date: 02 Dec. 2010
3 6 1
A = 1 2 2
2 4 5
1. (Null space) Find the null space of A if
Fall 2010
1 7
3 1.
8 4
2. (Span) Consider the vectors v1 = (0, 1
The Hong Kong University of Science and Technology
MATH 2350

Fall 2012
MATH 152
Applied Linear Algebra and Dierential Equations
Home Work 4
Matrix, System of Linear Equations
0
1. (Inverse) Verify that A = 0
1
Then compute the inverses of A2
Fall 2010
Due Date: 18 Nov. 2010
0 1
1 2
1 2 and B = 2 1
2 3
1
0
and At .
1
0 are in