MATH 246E, Sect.0901, Fall 1997: Differential Equations for Scientists and Engineers

There is also a page for all three MATH246E sections. Note that the problem sets there are different from Matlab assignments for this section.

Links marked with (PS) are Postscript files. If a postscript viewer is installed they can be viewed on the screen. They can also be saved to a file and printed out.

News

• Exam 3 is graded and available at my office.

• Exam 3 is on Thu, Dec. 11. It covers sections 3.8, 3.9, 6.2, 6.3, 7.5-7.7, 9.1. You can bring a calculator.

• The final exam is on Mon, Dec. 15 from 1:30pm to 3:30pm. It will be in MTH 0302. You can bring a calculator.

• Deadline for Assignment #3 extended to Fri, 9pm.

• Using Matlab for autonomous systems is now available

• New: Use tap matlab instead of tap matlab51 . To start Matlab, type tap matlab and matlab -nosplash .

• Deadline for Assignment 3 is extended to Thursday, Dec. 4

• Hints for Assignment 3
  • problem 1: How to change the plot style in ezplot

    In (a), plot only the curve for the natural frequency in a different style.

    In (b), plot only the curve for critical damping in a different style.

  • problem 2: You should use the procedure explained on the web page Solving ODEs using the Laplace Transform in Matlab, and not the procedure explained in Chapter 11 which was handed out in class

  • Problem 2(c): You don't have to plot f(t) since ezplot does not work with Dirac
  • Problem 3(a): There is a mistake in the text of problem 3(a).

    I want two plots:

    • The first plot (with a t-axis and a y-axis) should show the five curves y(t) vs. t (but no vector field which does not make sense here).
    • The second plot (with a y-axis and a y'-axis) should show the vector field and the five curves y'(t) vs. y(t). Here you should choose the range of the two axes so that all the curves are completely visible, and the same ranges should be chosen for the vector field.
  • Problem 3(b): I want two plots (both in the phase plane):
    • The first plot (with a y-axis and a y-axis) should show the curve y'(t) vs. y(t) (and nothing else). I recommend to choose the ranges [-1, 3.5] and [0, 2.5] for the two axes so that you can clearly see where the trajectory crosses the vertical axis. (To find a numerical value where this crossing is, inspect the entries of the vectors which you are plotting).
    • The second plot (with a y-axis and a y-axis) should show the five y'(t) vs. y(t) curves from (a), together with the y'(t) vs. y(t) curve from (b). The range of the axes should be chosen the same as in (a).
• Common Problems with Matlab is updated (Nov. 5)

• Using Matlab for Higher Order ODEs and Systems of ODEs is now available. Contents: Solving ODEs numerically and symbolically, plotting the solutions, phase plane plots, vector fields.

Course Outline

Information about time & place, instructor, textbooks, syllabus, grading policy, Matlab assignments, final exam. This was handed out in the first class.

Assignments

• Problem Set A (PS, 3 p.): Problem set for practicing Matlab. It wiill not be collected or graded. Must be completed by each student by Thu, Sept. 11.
  Solutions for Problem Set A: You should study this to see how to hand in homework problems (printouts of file prob.m, prob.txt and graphics for each problem).
• Assignment 1 was handed out on Sept. 18 and was due on Sept. 30.
• Assignment 2 was handed out on Oct. 23 and was due Fri, Nov. 7.
• Assignment 3 was handed out on Nov. 20 and is due on Thu, Dec. 4

Recommended Problems from the Textbook

These problems will be similar to problems of the exams and the final exam. But they will not be collected or graded. Note that partial solutions to the problems are in the back of the textbook.

Section 1.1 (Introduction), p.10

1-6

Section 2.1 (Linear equations), p. 23

only find the general solution: 1, 2, 6, 10
13, 14, 16, 18

Section 2.3 (Separable equations), p. 38:

1, 2, 3, 5, 6, 9, 10, 13, 14
7, 21, 22

Section 2.5 (Modeling with Linear Equations), p. 54:

1, 2, 3, 23, 24, 28

Section 2.6 (Population Dynamics and Related Problems), p. 69:

3, 4, 10

Section 2.8 (Exact Equations and Integrating Factors), p. 88:

2, 3, 7, 13, 19

Section 2.10 (Misc. Problems), p. 94:

1, 3, 5, 6, 8, 17, 20, 22

Section 8 (Numerical Methods)

problems

Section 3.1 (Homogeneous Equations with Constant Coefficients), p. 128

1, 5, 8, 9, 10

Section 3.4 (Complex Roots), p. 150

7, 9, 12, 17, 21

Section 3.5 (Repeated Roots), p. 159

1, 11, 12, 13, 14

Section 3.6 (Method of Undetermined Coefficients), p. 171

1, 3, 6, 7; only (a) for: 19, 20, 23

Section 3.7 (Variation of Parameters), p. 177

5, 7, 10

Section 3.8 (Mechanical and Electrical Vibrations), p. 190

1, 5, 11, 17
Note: weight is the force with which mass is pulled by earth's gravity. It is related to the mass m by w = m g , where g = 9.80665 m s^-2 = 32.174 ft s^-2.
``mass stretches spring a distance d'' means that spring constant k = w/d = m g/d

Section 3.9 (Forced Vibrations), p. 198

5, 7

Section 6.2 (Solution of IVP using the Laplace Transform), p. 303

(you can use Matlab for the inverse transform) 11, 21, 22, 23

Section 6.3 (Step Functions), p. 311

(do all these problems by hand without Matlab): 2, 7, 8, 16, 18

Section 6.4 (Discontinuous Forcing Functions), p. 318

(you can use Matlab for the inverse transform): 1, 5, 9

Section 6.5 (Impulse Funcions), p. 324

(you can use Matlab for the inverse transform): 1, 4, 5

Section 7.5 (Systems of First Order Linear Equation), p. 378

(in all problems, find the general solution and sketch a phase portrait): 2, 3, 5, 7

Section 7.6 (Complex Eigenvalues), p. 387

(in all problems, find the general solution and sketch a phase portrait): 1, 4, 6

Section 7.7 (Repeated Eigenvalues), p. 396

(in all problems, find the general solution and sketch a phase portrait): 2, 3

Section 9.1 (The Phase Plane: Linear Systems), p. 469

(do parts (a), (b), (c) in all problems): 1, 2, 5, 7, 10, 11

Section 9.3 (Almost Linear Systems), p. 488

(do parts (a), (b), (c) in all problems): 5, 6, 10, 12

Syllabus

BD indicates sections from the textbook (Boyce & DiPrima), M indicates sections from the book ``Differential Equations with Maple''.

already covered in class

• BD 1.1: Classification of differential equations
• BD 2.1, 2.2: Linear equations
• BD 2.3: Separable equations
• BD 2.4: Differences Between Linear and Nonlinear Equations
• BD 2.5: Modeling with Linear Equations
• BD 2.6: Population Dynamics and Related Problems
• M 5: Solution of Differential Equations, Matlab version (PS, 10 p.) available
• M 6: A qualitative approach to differential equations, Matlab version (PS, 9 p.) available
• (BD 2.7 was skipped)
• BD 2.8: Exact Equations and Integrating Factors
• BD 8.1, 8.2: Euler Method, Errors
• BD 8.3, 8.4: Improved Euler, Runge-Kutta methods
• M 7: Numerical Methods, Matlab version (PS, 16 p.) available
• BD 3.1, 3.4, 3.5: Homogeneous Equations with Constant Coefficients
  (reduction of order in 3.5 was not done)
  Summary of all cases is at bottom of p. 157
• BD 3.6: Nonhomogeneous Equations; Method of Undetermined Coefficients
• BD 3.7: Variation of Parameters
• M 9: 2nd Order Equations, Matlab version (PS, 6p.) available. Read only introduction, section ``Second Order Equations with Matlab''.
• BD 3.2, 3.3
• BD 3.8, 3.9
• BD 6.1-6.2; M 11 (Matlab version (PS, 10p.))
• BD 7.4, 7.5, 7.6, 7.7, M 12 (Matlab version (PS, 16 p.))
• BD 9.1-9.3, M 13 (Matlab version (PS, 8 p.))

  There is an error in Boyce/diPrima in table 9.3.1 on page 484 about types of critical points of almost linear systems:

  r₁ = r₂ > 0 or r₁ = r₂ < 0

  the type of the almost linear system should be N, SP or other.

  r₁ = i , r₂ = -i with nonzero

  the type of the almost linear system should be C, SP or other.

  Here is a correct table. One can construct examples where ``other'' types than the ones in table 9.3.1 occur.

Matlab Related Information for This Course

You are expected to read these documents carefully and try out the explained commands and examples on the computer. Items marked with ``*'' were handed out in class.

• *Introductory Handout (PS, 2 p.): How to get accounts, where to find computers, how to start Matlab, online help
• *Getting started with Matlab (PS, 10 p.): The most important Matlab commands for numerical and symbolic operations, and plotting (updated to explain the use of clear and Control-C)
• *Working with Matlab (PS, 3 p.): How to use m-files, how to hand in your homework (updated with new instructions for script m-files: put ``clear'' and ``close all'' at the beginning, do not put ``diary'' in m-file)
• Using Matlab for First Order ODEs: How to plot direction fields, solve ODEs numerically and symbolically, plot solutions
• Using Matlab for Higher Order ODEs and Systems of ODEs: Solving ODEs numerically and symbolically, plotting the solutions, phase plane plots, vector fields.
• Solving ODEs using the Laplace Transform in Matlab
• Using Matlab for Autonomous Systems
• Common Problems with Matlab: If you have trouble check this page before asking the tutor.

Matlab Version of ``Differential Equations with Maple''

Selected chapters are available here. Items marked with ``*'' were handed out in class.

• *Solutions of Differential Equations (PS, 10p.) (Chapter 5) explains how to solve differential equations symbolically with Matlab. It also explains existence and ``stability'' of solutions of differential equations.
• *Qualitative Approach to Differential Equations (PS, 9p.) (Chapter 6) explains how to plot direction fields with Matlab. It also discusses autonomous equations, critical points and their stability.
• *Numerical Methods (PS, 16 p.) (Chapter 7). Contents: Numerical solution ODEs using ode45 in Matlab; Euler, improved Euler and Runge-Kutta methods; controlling the eror in ode45, reliability of numerical methods.
• *Qualitative Theory of Second Order Linear Equations (PS,6 p.) (Chapter 9). It does not contain all of Chapter 9 in the book. You only have to read the introduction and the section ``Second Order Equations with Matlab''.
• *Laplace Transforms (PS, 10 p.) (Chapter 11)
• Higher Order Equations and Systems of First Order Equations (PS, 16 p.) (Chapter 12)
• Qualitative Theory for Systems of Differential Equations (PS, 8 p.) (Chapter 13)

Other Computer Information

• Matlab Primer (PS, 35 p.): Explains basic Matlab commands (but not symbolic operations, differential equation commands)
• Matlab Help Pages (accessible from Matlab by typing helpdesk)
• How to get a WAM account, GLUE account, Printing account
• Text editors (for writing m-files): Pico, vi, emacs
• Getting started with your WAM Unix account
• How to print text files for free

Tutoring with Matlab

A tutor (Ali Hirsa) will help students in computer labs with Matlab. He will not do the problem sets for you, but only help with technical questions about the computers and Matlab. Before you ask the tutor check the page Common Problems with Matlab.

The tutoring is in CSS 4352. The schedule is as follows:

Tobias von Petersdorff , tvp@math.umd.edu