The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Summer 2015
MATH 2352
Dierential Equations and Applications
Practical Exercise 2
Secondorder Dierential Equations
Suggested Solution
1. (Linear property for secondorder linear dierential equations)
(a) If y1 (t) and y2 (t) are two solutions of the linear homogeneou
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Fall 2012
MATH 2352 Solution Sheet 06
by Xiaoyu Wei
Created on April 17, 2015
[Problems] 5.5: 6, 11; 5.6: 10, 18;
5.5  6. For the equation
x2 y + x y + (x 2) y = 0.
(a) Show that the given differential equation has a regular singular point at x = 0.
(b) Determine
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Fall 2012
MATH 2352 Solution Sheet 07
by Xiaoyu Wei
Created on April 19, 2015
[Problems] 6.1: 5(c), 14; 6.2: 2, 8, 14, 21; 6.3: 9, 11, 13, 20, 23, 25;
6.1  5. (c) Find the Laplace transform of each of the following functions
f (t) = tn ;
where n is a positive inte
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Fall 2012
MATH 2352 Solution Sheet 03
by Xiaoyu Wei
Created on March 6, 2015
[Problems] 2.7: 16; 3.1: 4, 11; 3.3: 14, 19; 3.4: 6, 11; 3.2: 9, 10, 14, 25; 3.5: 6, 20;
2.7  16. Consider the initial value problem
y = t2 + y 2,
y(0) = 1.
Use Eulers method with h = 0.1
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Fall 2012
MATH 2352 Solution Sheet 02
by Xiaoyu Wei
Created on February 24, 2015
[Problems] 2.2: 3, 7, 16(a,c); 2.3: 2, 9; 2.4: 4, 6, 13; 2.6: 7, 9
2.2  3. Solve the given differential equation
y + y 2 cos x = 0.
Solution. Multiply both sides by y 2, we have
y
+ c
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Fall 2012
MATH 2352 Solution Sheet 08
by Xiaoyu Wei
Created on April 20, 2015
[Problems] 6.4: 10, 12; 6.5: 3, 7; 6.6: 1(c), 7, 10, 17, 20;
6.4  10. For
5
y 00 + y 0 + y = g(t);
4
y(0) = 1;
y 0(0) = 0;
g(t) =
06t<
:
t>
sin t;
0;
(a) Find the solution of the given i
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Fall 2012
MATH 2352 Solution Sheet 10
by Xiaoyu Wei
Created on May 24, 2015
[Problems] 7.7: 6, 12, 13; 7.8: 7, 18, 22 7.9: 1, 3.
7.7  6. For
x =
0
1 1
4 1
x;
(a) Find a fundamental matrix for the given system of equations.
(b) Also nd the fundamental matrix (t) sa
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Fall 2012
MATH 2352 Solution Sheet 11
by Xiaoyu Wei
Created on May 24, 2015
[Problems] 8.1: 12(a,c), 23; 8.2: 4(b), 22; 8.3: 1(b).
8.1  12. For the initial value problem
y 0 = (y 2 + 2ty)/(3 + t2);
y(0) = 0.5;
nd the approximate values of the solution at t = 0.5;
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Fall 2012
MATH 2352 Solution Sheet 09
by Xiaoyu Wei
Created on May 24, 2015
[Problems] 7.1: 5, 6, 8; 7.4: 4, 9; 7.5: 15, 17, 19; 7.6: 7, 15(a, b);
7.1  5. Transform the given eqaution into a system of rst order equations.
u 00 + 2u 0 + 4u = 2 cos 3t;
u(0) = 1;
u 0
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Fall 2012
MATH 2352 Solution Sheet 01
by Xiaoyu Wei
1.2: 2c, 3, 7
1.3: 16, 18, 20, 2124
2.1: 6, 8, 16
1.2  2  c. Solve the following initial value problems and plot the solutions for several values
of y0.
dy /dt=2y 6,
y(0)= y0.
Solution. y(t) = y0 e2t + 3.
1.2
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Summer 2015
MATH 2352
Dierential Equations and Appplications
Practical Exercise 4
Laplace Transform
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 show that L
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Summer 2015
MATH 2352
Dierential Equations and Appplications
Practical Exercise 5
System of Dierential Equations
Suggested Solution
1. (Dierential system distinct real eigenvalues) Solve the dierential system
x (t) =
Solution
5
4
2
x(t).
1
Let A be the coecient matri
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Summer 2015
MATH 2352  Dierential Equations and Applications
Part 3  Series Solutions for SecondOrder Linear Equations
1. Brief Review of Power Series
1.1 Innite Series
# Denition:
ak is called an
For an innite set of numbers cfw_a1 , a2 , a3 , . . ., their sum S
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Summer 2015
MATH 2352  Dierential Equations and Applications
Part 5  Systems of FirstOrder Linear ODEs
1. Eigenvalues and Eigenvectors
Consider the multiplication of matrices A =
AX =
2
0
0
3
x1
x2
=
2x1
3x2
2
0
0
3
,X=
x1
x2
,
.
The matrix A can be considered as
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Summer 2015
MATH 2352  Dierential Equations and Applications
Part 2  SecondOrder Linear Ordinary Dierential Equations
1. Introduction
y = f (t, y, y ) .
General form of a 2ndorder ODE:
It is linear if can be written as
y + p(t)y + q(t)y = g(t) .
Otherwise, it is
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Summer 2015
MATH 2352  Dierential Equations and Applications
Part 4  Laplace Transform
1. Introduction
Consider the initial value problem (IVP) of a springmass system with the governing equations given by
k
u + m u + m u = f (t)
.
u(0) = u0 , u (0) = u0
We may obt
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Summer 2015
MATH 2352
Home Work 4
Dierential Equations and Applications
Laplace Transform
1. Solve the initial value problem by the method of Laplace Transform
y + y = cos 2t
, y(0) = 1, y (0) = 0 .
Suggested Solution
2. Solve the initial value problem by the method
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Summer 2015
MATH 2352
Home Work 2
Dierential Equations and Applications
Secondorder Dierential Equations
Suggested Solution
1. By the method of undetermined coecients, nd the general solution of the following dierential
equation
u + 9u = 2x2 e3x + 5 .
2. Given that
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Summer 2015
MATH 2352
Home Work 1
Dierential Equations and Applications
Firstorder Dierential Equations
1. Solve the initial value problem
dr
+ r tan = cos2 ,
d
r( ) = 1 .
4
Suggested Solution
2. Consider the equation
ty (4 y)
dy
=
.
dt
1+t
(a) Find the general solu
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Summer 2015
MATH 2352
Home Work 3
Dierential Equations and Applications
Power Series Methods
1. Use the power series method centered at t0 = 0 to solve
1 + t2 y + 4ty + 2y = 0 ,
and determine the radius of convergence of the power series solution.
Suggested Solution
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Summer 2015
MATH 2352
Dierential Equations and Applications
Practical Exercise 3
Power Series Methods
Suggested Solution
1. (Ordinary point) For the dierential equation y xy y = 0, determine the recurrence
relation for the terms of the power series solution about the
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Summer 2015
MATH 2352
Dierential Equations and Applications
Practical Exercise 1
Suggested Solution
Firstorder Dierential Equations
1. (Linear vs nonlinear) Indicate whether the following equation is linear or nonlinear.
1.
dy
sin t = t2 y
dt
7. yy = t3 + y sin 3t
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Fall 2012
Math 2352Spring 2016
Worksheet 4: Inhomogeneous odes
To receive credit, hand in as many solved practice problems as time permits. Try unfinished
problems at home. Solution of this worksheet will be made available on the website.
1. (Demonstration) Explain
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Fall 2012
Math 2352Spring 2016
Week 03 Worksheet: Secondorder homogeneous odes
To receive credit, hand in as many solved practice problems as time permits. Try unfinished
problems at home. Solution of this worksheet will be made available on the website.
1. (Demon
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Fall 2012
MATH 2352 Problem Sheet 03
by Xiaoyu Wei
Created on March 3, 2015
[Problems] 2.7: 16; 3.1: 4, 11; 3.3: 14, 19; 3.4: 6, 11; 3.2: 9, 10, 14, 25; 3.5: 6, 20;
2.7  16. Consider the initial value problem
y = t2 + y 2,
y(0) = 1.
Use Eulers method with h = 0.1,
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Fall 2012
 CHAPTER 8. 
Chapter Eight
Section 8.1 2. The Euler formula for this problem is C8" oe C8 2^& >8 $C8 , C8" oe C8 &82# $2 C8 ,
in which >8 oe >! 82 Since >! oe ! , we can also write
a+b. Euler method with 2 oe !& >8 C8 8oe# !" "&*)! 8oe% !# "#*#) 8oe' !$
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Fall 2012
 CHAPTER 6. 
Chapter Six
Section 6.1 3.
The function 0 a>b is continuous. 4.
The function 0 a>b has a jump discontinuity at > oe " . 7. Integration is a linear operation. It follows that (
E !
9=2 ,> /=> .> oe
" E ,> => " E ,> => ( / / .> ( / / .> # !
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Fall 2012
 CHAPTER 5. 
Chapter Five
Section 5.1 1. Apply the ratio test : lim aB $b8" k a B $b 8 k
Hence the series converges absolutely for kB $k " . The radius of convergence is 3 oe " . The series diverges for B oe # and B oe % , since the nth term does not a
The Hong Kong University of Science and Technology
Differential Equations and Applications
MATH 2352

Fall 2012
 CHAPTER 4. 
Chapter Four
Section 4.1 1. The differential equation is in standard form. Its coefficients, as well as the function 1a>b oe > , are continuous everywhere. Hence solutions are valid on the entire real line. 3. Writing the equation in standa