HOMEWORK (MATH 319 - LECTURE 3 - FALL 2014) Introduction

HOMEWORK (MATH 319 - LECTURE 3 - FALL 2014)
Introduction
• Each Thursday homework problems will be assigned and will be collected following Thursday in the class room. Graded HW will be returned at your discussion
section.
• Students are encouraged to collaborate on assignment problems in groups;
however, submitted assignments must be the work of the individual submitting the
assignment. Copying any portion of an assignment directly is academic fraud!
• Late assignments will not be graded.
• Print the cover page, fill out your name, student ID, and discussion section,
and affix it to the beginning of your assignment firmly with staple.
1. First week
1.1. Class #1 (Sep. 2). We start with Ex.1 and Ex.2 in Section1.1. From Ex.1
we illustrate the way to write down a differential equation from physical model. In
Ex.2 we see an example of a direction field. Solutions of such DE are given in
Ex.2, Sec.1.2 and we will learn the concept of the initial value problem. In Sec.1.3,
we learn the difference of ODE and PDE and what solutions mean.
The subject of Ch.2 is the first order ODE dy
dt = f (t, y). In Sec. 2.1, we consider
dy
the linear first order ODE: dt + p(t)y = g(t). We learn the way (method of
integrating factors) to write down solutions to such ODE (pp 36–37). Then using
this method, we solve Ex.2.
1.2. Class #2 (Sep. 4). We review the method of integrating factors. Then we
solves several problems with this method. Then we move to Sec.2.2.: separable
equations.
1.3. HW #1 (Due day Sep. 11). : complete problem set
Problem 1.
(a) Draw a direction field of y 0 + 3y = te−3t
(b) Based on an inspection of the direction field of y 0 + 3y = te−3t , describe how
solutions behave for large t.
(c) Find the solution of
y 0 + 3y = te−3t ,
Problem
Problem
Problem
Problem
Problem
2.
3.
4.
5.
6.
y(1) = 0.
Section 2.1. Problem 25
Section 2.1. Problem 33
Solve the initial value problem ty 0 − y 2 = 1, y(1) = 1.
Section 2.2. Problem 7
Section 2.2. Problem 22
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2. Second week
2.1. Class #3 (Sep. 9). We review both previous sections with two problems.
Then move to section 2.3. Solve example 3, example 4, problem 17 with (i).
2.2. Class #4 (Sep. 11). We solve problem 17 with (i). We finish section 2.4
and move to section 2.5.
2.3. HW #2 (Due day Sep. 18). : complete problem set
Problem 1. Heat transfer from a body to its surroundings by radiation, based on
the Stefan-Boltzmann law, is described by the DE
du
= −α(u4 − T 4 ),
dt
where u(t) is the absolute temperature of the body at time t, T is the absolute
temperature of the surroundings, and α is a constant depending on the physical
parameters of the body.
Suppose that a body with initial temperature 2000K is surrounded by a medium
with temperature 300K and that α = 2.0 × 10−12 K −3 /s.
(a) Determine the temperature of the body at any time by solving the ODE.
Hint 1 : You may determine A, B, C, and D satisfying
A
B
Cu + D
1
=
+
+ 2
u4 − T 4
u+T
u−T
u + T2
Hint 2 : You may use the change of variables u = T tan θ to evaluate
Z
1
du.
u2 + T 2
(b) Draw by hand a graph of u versus t.
Problem 2. Consider an insulated box with internal temperature u(t). According
to Newton’s law of cooling, u satisfies the DE
du
= −k[u − T (t)],
dt
where T (t) is the ambient temperature. Suppose that T (t) = T0 + T1 sin ωt.
(a) Solve this DE. Observe that part of your solution approaches zero as t becomes large; this is called the transient part. The remainder of the solution is called
the steady state; denote it by S(t).
(b) Suppose that t is measured in hours and what ω = π/12, corresponding to a
period of 24 h for T (t). Further, let T0 = 15o C, T1 = −10o C, and k = 0.2/h. Draw
graphs of S(t) and T (t) versus t on the same axes. From your graph estimate the
amplitude R of the oscillatory part of S(t). Also estimate the time lag τ between
corresponding maxima of T (t) and S(t).
Problem 3. Section 2.3. Problem 27.
Problem 4. Solve the initial value problem
1 + 3t2
dy
= 2
, y(0) = 3.
dt
3y − 12y
and determine the interval in which the solution is valid. Explain why this solution
is unique (using Theorem 2.4.2 in page 70)
Problem 5. Section 2.4 Problem 22
2
Problem 6. Section 2.4 Problem 23
3. Third week
3.1. Class #5 (Sep. 16). 2.5. Autonomous Equation, 2.6 Exact Equations
3.2. Class #6 (Sep. 18).
3.3. HW #3 (Due day Sep. 25). complete problem set
Problem 1. Section 2.5. Problem 26 and an extra problem:
3
(d) Solve dy
dt = ay − y . (You may split the cases a < 0, a = 0, and a > 0. )
Problem 2. Section 2.5. Problem 21 and an extra problem:
(f) Solve dy
dt = r(1 − y/K)y − h.
Problem 3. Determine whether the following equation is exact.
(yety cos 3t − 3ety sin 3t + 2t) + (tety cos 3t − 3)y 0 = 0.
If it is exact, find the solution.
Problem 4. Section 2.6. Problem 32
Problem 5. Solve
(6t2 y + 4ty + y 3 ) + (2t2 + y 2 )y 0 = 0.
Problem 6. Section 2.5. √Problem 19 and an extra problem:
(c) Solve dh
dt = (k − αa 2gh)/A.
4. 4th week
4.1. Class #7 (Sep. 23). review 2.6 Exact Equations, Section 3.1 to Section 3.4.
4.2. Class #8 (Sep. 25).
4.3. HW #4 (Due day Oct. 2). complete problem set
Problem 1. Write down the general solution of y 00 + 6y 0 + 8y = 0.
Problem 2. Determine the longest interval in which the given initial value problem
is certain to have a unique twice-differentiable solution. Copy the proper Theorem
to justify your answer. Do not attempt to find the solution.
(t − 1)(sin t)y 00 + t2 y 0 + 5y = 1,
Problem
Problem
Problem
Problem
Problem
Problem
3.
4.
5.
6.
7.
8.
Section
Section
Section
Section
Section
Section
3.2.
3.3.
3.3.
3.3.
3.4.
3.4.
Problem
Problem
Problem
Problem
Problem
Problem
y(2) = 0, y 0 (2) = 1.
26
22
24
28
11
26
5. 5th week
5.1. Class #9 (Sep. 30), Class #10 (Oct. 2). Section 3.5 to Section 3.8
5.2. No HW. : There is no HW for the 5th week due to Mid term exam (9th Oct).
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6. 6th week
6.1. Class #11 (Oct. 7), Class #12 (Oct. 9).
6.2. HW #5 (Due day Oct. 16). : Complete set, These problems will be solved
in the class (Oct 14).
Problem 1. Section 3.7. Problem 9
Problem 2. Section 3.8. Problem 11
Problem 3. Section 3.8. Problem 17
7. 7th week
7.1. Class #13 (Oct. 14), Class #14 (Oct. 16). Finishing Chapter 3 (Sec 3.7
and Sec 3.8) and start Chapter 5 (Sec 5.1, Sec 5.2)
7.2. HW #6 (Due day Oct. 23). : Complete set,
Problem 1.Section 5.1. Problem 7
Problem 2.Section 5.1. Problem 25
Problem 3.Section 5.1. Problem 28
Problem 4.Section 5.2. Problem 4
Problem 5.Section 5.2. Problem 6
8. 7th week
8.1. Class #15 (Oct. 21), Class #16 (Oct. 23).
8.2. HW #7 (Due day Oct. 30). : Complete set
Problem 1.Section 5.2. Problem 8
Problem 2.Section 5.2. Problem 21
Problem 3.Section 5.2. Problem 22
Problem 4.Section 5.3. Problem 2
Problem 5.Section 5.3. Problem 3
Problem 6.Section 5.3. Problem 6
Problem 7. Following Example 1, Example 2, Example 3 in Section 5.4, solve
(1) 2x2 y 00 + 5xy 0 + y = 0, x > 0.
(2) x2 y 00 + 7xy 0 + 9y = 0, x > 0.
(3) x2 y 00 + xy 0 + 4y = 0, x > 0.
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