MATH 1851 - Ordinary Differential Equations (ODEs)
(A)
First order equations
A linear, first order ODE is always solvable by the method of integrating factor:
dy
p ( x) y q ( x) .
dx
Multiplying by (the integrating factor)
exp
p( x) dx
will turn the le
MATH1851 (13/14 Semester 2) Assignment 1 (0%)
(Due date: 14 Feb 2014 5pm)
06-02-2014
Please submit your solution to the assignment box located on the 4th oor
of Run Run Shaw building. DO NOT COPY. Late submission will not be
accepted.
1. (Ch1 A13) If limx
Solutions to Supplementary Exercise
Section
Questions
9.4
19,21,26
9.5
11,14
9.6
10,13,15,22,26
(Ex9.4 Q19)
The equation is exact. Write M = 2x +
F =
y
,N
1+x2 y 2
=
x
1+x2 y 2
2y. Then
M dx = x2 + tan1 xy + g(y)
and
N=
x
F
=
+ g (y).
y
1 + x2 y 2
Takin
Solutions to Supplementary Exercise
Section Questions
10.2
9,16,27,28
10.3
22
10.4
15,16
10.5
17,19,20
(Ex10.2 Q9)
The auxilliary eqaution is 4r2 4r + 1 = 0, which has a double root r = 1/2. Hence
the general solution is
y = C1 et/2 + C2 tet/2 .
(Ex10.2
Solutions to Supplementary Exercise
Section
Questions
10.6
4,11
10.7
12,15,37,40,43
(Ex10.6 Q4)
A pair of linearly independent solutions to the homogeneous equation are
y1 = et , y2 = tet .
Then a particular solution to the nonhomogeneous equation is giv
MATH 1851 - Laplace Transforms
(A)
Basic concepts
The Laplace transform of a function f (t) is defined by
L ( f (t ) ) L( f )
0
e st f (t ) dt F ( s) ,
f (t ) L1 ( F (s),
where t is usually time. The convention here is that small letter is used for the f
MATH1851 Laplace Transforms Tutorial Ex
1.
Find the Laplace
t
f (t) = 20 t
0
transform of the function from denition:
0 < t < 10
10 < t < 20
t > 20
[Ans:
(1e10s )2
]
s2
2.
Given sinh x = 1 (ex ex ), cosh x = 1 (ex + ex ), nd L(cosh at + k sinh at).
2
2
[A
Chapter 5. Integration
Topics
1. Area and Estimating with Finite Sums
2. Definite Integrals
3. The Fundamental Theorem of Calculus
4. Definition of Natural Logarithms
5. Interlude Hyperbolic Functions
HKU MATH1851
1. Area and Estimating with Finite Sums
2
Chapter 3. Applications of derivatives
Topics
1. Extremum points
2. The Mean Value Theorem & its consequences
3. Monotonicity and the First Derivative
4. Concavity and the Second Derivative
HKU MATH1851
1. Extremum Points
2
3
4
5
<=
6
7
2. The Mean Value
MATH 1851 Quiz 2, April 2015
Answer all questions
NAME (in English and if applicable, in Chinese):
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UNIVERSITY NUMBER:
gimme/2 KW
Questibnm Markﬂsﬂ i
1 W ‘ r l
2 — — . m «H
L Ms -ﬂ
3
. m w g
5 l
.6 a m l
:TOTAL~
I .
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1851: CALCULUS AND ORDINARY DIFFERENTIAL EQUATIONS
2:30p.m.
15 May, 2014
~
5:30p.m.
No calculator can be used in this examination.
Answer ALL TEN questions
Please Use Separate Answer Books for Ques
Chapter 8. Techniques of Integration
Topics
1. Integration by Parts
2. Method of Substitutions
3. Trigonometric Substitutions
4. Trigonometric Integrals
5. Resolve Rational Functions into Partial Fractions
HKU MATH1851
1. Integration by Parts
2
3
4
2. Met
1. Rate of change & Tangents
Chapter 1. Limits and Continuity
Topics
1. Rate of change & Tangents
2. Limit of a Function and Limit Laws
3. The precise definition of a limit
4. One-sided limits
5. Continuity
6. Limits involving infinity
HKU MATH1851
2
3
4
Chapter 2. Differentiation
Topics
1. Tangents and the derivative at a point
2. Differentiation formulas and rules
3. The Chain rule
4. Implicit differentiation
5. Differentiation formulas for natural logarithms & exponential functions
6. Logarithmic Diffe
Chapter 3. Applications of derivatives
Topics
1. Extremum points
2. The Mean Value Theorem & its consequences
3. Monotonicity and the First Derivative
4. Concavity and the Second Derivative
HKU MATH1851
1. Extremum Points
2
3
4
5
6
7
2. The Mean Value The
Chapter 5. Integration
Topics
1. Area and Estimating with Finite Sums
2. Definite Integrals
3. The Fundamental Theorem of Calculus
4. Definition of Natural Logarithms
5. Interlude Hyperbolic Functions
HKU MATH1851
1. Area and Estimating with Finite Sums
2
Chapter 6. Applications of Definite
Integrals
1.
2.
3.
4.
5.
6.
Volumes using Cross-Sections
Solids of Revolution: The Disk Method
Solids of Revolution: The Washer Method
Volume using Cylindrical Shells
Arc Length
Area Surfaces of Revolution
HKU MATH1851
Chapter 2. Differentiation
Topics
1. Tangents and the derivative at a point
2. Differentiation formulas and rules
3. The Chain rule
4. Implicit differentiation
5. Differentiation formulas for natural logarithms & exponential functions
6. Logarithmic Diffe
Chapter 8. Techniques of Integration
Topics
1. Integration by Parts
2. Method of Substitutions
3. Trigonometric Substitutions
4. Trigonometric Integrals
5. Resolve Rational Functions into Partial Fractions
HKU MATH1851
1. Integration by Parts
2
3
4
2. Met
Chapter 7
Transcendental Functions
The number in [ ] refers to the page number of our textbook
7.5 Indeterminate Forms and L Hpitals Rule
o
Theorem 5 - L Hpitals Rule [377] Suppose that f (a) = g(a) = 0, that f and g are
o
dierentiable on an open interval
Q1. Find the average rate of change of the
function dened by P () = 3 42 + 5 on
the interval [1, 2].
Answer. Average rate of change (check denition) is
change in P ()
P/ =
change in
which is
P (2) P (1)
=0 .
21
1
Q2. Find the tangent line to the curve y
MATH 1851 - Laplace Transforms
(A)
Basic concepts
The Laplace transform of a function f (t) is defined by
L ( f (t ) ) = L( f ) =
0
e st f (t ) dt = F ( s) ,
f (t ) = L1 ( F (s),
where t is usually time. The convention here is that small letter is used f
Chapter 1: Limits and Continuity
Chapter 1. Limits and Continuity
Topics
1. Rate of change & Tangents
2. Limit of a Function and Limit Laws
3. The precise definition of a limit
4. One-sided limits
5. Continuity
6. Limits involving infinity
HKU MATH1851
1.
MATH 1851 - Ordinary Differential Equations (ODEs)
(A)
First order equations
Linear first order equations
A linear, first order ODE is always solvable by the method of integrating factor. It will
first be convenient to write the equation in a canonical fo
Ordinary differential equations
Professor Kwok Wing CHOW
Mechanical Engineering, HKU
Motivation There are many situations in
science and engineering where a
quantitative relation exists between a
function and its rate of change. Examples
are
(a) Populatio
Laplace transforms
Professor K. W. Chow
Mechanical Engineering, HKU
Physical Motivation:
To look at the dependent variable
from another physical perspective,
e.g. displacement of a string as a
function of space and time
converted to a frequency or wave
le
MATH1851 Calculus and Ordinary Differential Equations
Ordinary Differential Equations: March May 2016
Professor Kwok Wing CHOW (teacher of the morning class, class C)
Room 7-18A, Haking Wong Building, HKU; 2859 2641; kwchow@hku.hk
Consultation hours: Mon