The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Math 2351Fall 2016
Week 10 Worksheet: Systems of equations
To receive credit, hand in as many solved practice problems as time permits. Try unfinished
problems at home. Solution of this worksheet will
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
WEEK HOMEWORK 5
(1) Solve the inhomogeneous ode using the Laplace transform technique: (10 pts)
x
+ 2x + 5x = et ,
x(0) = 0, x(0)
= 0.
(2) Use Laplace transform to find a particular solution for (12p
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
WEEK HOMEWORK 6
(1) Find two independent power series solutions to the following differential equation, computing
solutions up to x5 .
[12pts]
y 00 + xy 0 + y = 0.
1
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
WEEK HOMEWORK 9
(1) Find all the fixed points of the following odes and classify their stability:
(a) x = x x3
(b) x = ex sin x
(c) x = 1 2 cos x
2
(d) x = 1 ex
(2) Find all the fixed points of x = x(
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
WEEK HOMEWORK 7
(1) Find the solution y = y(x) of the following initial value problems.
x2 y 00 2xy 0 + 2y = 0,
x > 0, y(1) = 0 , y(2) = 1.
(2) Find the general solution and sketch the phase space dia
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
MATH 2351: Additional Explanations and Hints to
Week 11 Tutorial
by li haohan
HKUST
Abstract
This document is provided as a part of supplemental materials for MATH 2351 Introduction
to Dierential Equa
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
MATH 2351: Additional Explanations and Hints to
Week 12 Tutorial
by li haohan
HKUST
Abstract
This document is provided as a part of supplemental materials for MATH 2351 Introduction
to Dierential Equa
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
MATH 2351: Additional Explanations and Hints to
Week 6 Tutorial
by li haohan
HKUST
Abstract
This document is provided as a part of supplemental materials for MATH 2351 Introduction
to Dierential Equat
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
MATH 2351: Additional Explanations and Hints to
Week 13 Tutorial
by li haohan
HKUST
Abstract
This document is provided as a part of supplemental materials for MATH 2351 Introduction
to Dierential Equa
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
MATH 2351: Additional Explanations and Hints to
Week 8 Tutorial
by li haohan
HKUST
Abstract
This document is provided as a part of supplemental materials for MATH 2351 Introduction
to Dierential Equat
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
WEEK 4HOMEWORK
(1) Solve the initial value problem:(9 pts)
x
+ 3x + 2x = e2t , x(0) = 0, x(0)
= 0.
(2) Find a particular solution for: (6pts)
x
+ x + x = sin 2t.
1
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
WEEK HOMEWORK 11
(1) Consider the general solution to a partial differential equation given by
X
1
u(r, ) = A0 +
rn (An cos n + Bn sin n).
2
n=1
Suppose that when r = a, one has the boundary condition
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Math 2351Fall 2016
Week 07 Worksheet: The Laplace transform
To receive credit, hand in as many solved practice problems as time permits. Try unfinished
problems at home. Solution of this worksheet wil
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Math 2351Fall 2016
Week 08 Worksheet: Series solutions
To receive credit, hand in as many solved practice problems as time permits. Try unfinished
problems at home. Solution of this worksheet will be
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
MATH 2351: Additional Explanations and Hints to
Week 1 Tutorial
by li haohan
HKUST
Abstract
This document is provided as a part of supplemental materials for MATH 2351 Introduction
to Dierential Equat
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
MATH 2351: Additional Explanations and Hints to
Week 5 Tutorial
by li haohan
HKUST
Abstract
This document is provided as a part of supplemental materials for MATH 2351 Introduction
to Dierential Equat
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
MATH 2351: Additional Explanations and Hints to
Week 4 Tutorial
by li haohan
HKUST
Abstract
This document is provided as a part of supplemental materials for MATH 2351 Introduction
to Dierential Equat
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
MATH 2351: Additional Explanations and Hints to
Week 2 Tutorial
by li haohan
HKUST
Abstract
This document is provided as a part of supplemental materials for MATH 2351 Introduction
to Dierential Equat
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Solutions to worksheet 3: Q16Q21
Q16
Let S(t) be the amount of money at t,
S(t + 4t) = S(t) + rS(t)4t k4t
S(t + 4t) S(t)
= rS(t) k
4t
S 0 (t) = rS(t) k
Solving for S(t):
k
S(t) = S0 ert ert (1 ert )
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
MATH 2351: Additional Explanations and Hints to
Week 3 Tutorial
by li haohan
HKUST
Abstract
This document is provided as a part of supplemental materials for MATH 2351 Introduction
to Dierential Equat
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
WEEK HOMEWORK 7
(1) Find the general solution and sketch the phase space diagrams of
x 1 = x1 + x2 ,
[13pts]
x 2 = 4x1 3x2 .
(2) Let a be a real number. Find the general solution of the system.
[13pts
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
MATH 2351: Additional Explanations and Hints to
Week 9 Tutorial
by li haohan
HKUST
Abstract
This document is provided as a part of supplemental materials for MATH 2351 Introduction
to Dierential Equat
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
MATH 2351: Additional Explanations and Hints to
Week 10 Tutorial
by li haohan
HKUST
Abstract
This document is provided as a part of supplemental materials for MATH 2351 Introduction
to Dierential Equa
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
MATH 2351: Additional Explanations and Hints to
Week 7 Tutorial
by li haohan
HKUST
Abstract
This document is provided as a part of supplemental materials for MATH 2351 Introduction
to Dierential Equat
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Math 2351 ODE and Application Tutorial 3 (1st Order ODE)
Page  1
Homogeneous Equation
Q6) Example 2.29
Q1 ) Example 2.21
The following problem is not exact but becomes
exact when multiplied by the gi
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Full Solutions to Math 2351 ODE and Application (Tutorial 5)
Q1
Q2
Page  1
Full Solutions to Math 2351 ODE and Application (Tutorial 5)
Q3
Q4
Page  2
Full Solutions to Math 2351 ODE and Application
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Full Solutions to Math 2351 ODE and Application (Tutorial 4)
Q1
Q2
Q3
Page  1
Full Solutions to Math 2351 ODE and Application (Tutorial 4)
Q4
Q5
Page  2
Full Solutions to Math 2351 ODE and Applicati
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
#8167 42124 Order om am. Q?) I (Pa)
3 7) yawn #2056 kW ye; are gowagepem MM 0342665
1) 94.74am! rm 7 . E
, , 3 OIQE. 0+ I y .
x W + #60 g] + CxJ : g0), M 66) K 1404 y? 7: a zamfaczrygfgj; 602.66 td
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Questions for Math 2351 ODE (Tutorial 4)
1)
Page  1
Example 3.1 (Fundamental Solution)
1
Verify that the functions 1 () = 2 and 2 () =
are solutions of
2
2 = 0,
> 0.
(3.1)
Wronskian and Linearly I
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
Question and Solutions to Math 2351 Classwork of Tutorial 4 (Class work 2) Page  1
Name : _
Time Allowed : 10 minutes
Student number : _
Q1) The method of undetermined coefficients
Find the form of a
The Hong Kong University of Science and Technology
Introduction to Differential Equations
MATH 2351

Fall 2013
MATH2351 Introduction to Differential Equations, 201718 Spring
Tutorial Worksheet 1: Introduction/Direction Fields
Name:
ID No.:
(T1b)
Tutorial Section:
Complete at least TWO questions from the follo