MAT2705 Differential Equations with Linear Algebra

Course Goals and Objectives

Elementary use of MAPLE is a required supporting tool in the entire MAT1500-1505-2500-2705 sequence of Calculus and Differential Equations with Linear Algebra for Science and Engineering majors.

Text: Differential Equations and Linear Algebra
Edwards and Penney Edition 2e 2004 Prentice Hall
ISBN: 0-13-148146-0
Textbook web page: http://vig.prenhall.com/catalog/academic/product/0,4096,0131481460,00.html and accompanying website: http://wps.prenhall.com/esm_edwards_dela_2

Chapters 1-7. The following core sections are recommended; optional sections are indicated with the square bracket notation [...], omitted sections with double square bracket notation [[..]]:

• 1. First-Order Differential Equations.
  • 1.1: Differential Equations and Mathematical Models.
  • 1.2: Integrals as General and Particular Solutions.
  • 1.3: Slope Fields and Solution Curves.
  • 1.4: Separable Equations and Applications.
  • 1.5: Linear First-Order Equations.
  • 1.6: [[Substitution Methods and Exact Equations.]]
• 2. Mathematical Models and Numerical Methods.
  • 2.1: Population Models.
  • 2.2: [[Equilibrium Solutions and Stability.]]
  • 2.3: [Acceleration-Velocity Models.]
  • 2.4: [Numerical Approximation: Euler's Method.]?
  • 2.5: [[A Closer Look at the Euler Method.]]
  • 2.6: [[The Runge-Kutta Method.]]
• 3. Linear Systems and Matrices.
  • 3.1: Introduction to Linear Systems.
  • 3.2: Matrices and Gaussian Elimination.
  • 3.3: Reduced Row-Echelon Matrices.
  • 3.4: Matrix Operations.
  • 3.5: Inverses of Matrices.
  • 3.6: Determinants. (emphasis on row operation evaluation; Cramer's rule and adjoint formula for inverse can be omitted after mentioning)
  • 3.7: [Linear Equations and Curve Fitting.]
• 4. Vector Spaces.
  • 4.1: The Vector Space R^3.
  • 4.2: The Vector Space R^n and Subspaces.
  • 4.3: Linear Combinations and Independence of Vectors.
  • 4.4: Bases and Dimension for Vector Spaces.
  • 4.5: [[Row and Column Spaces]]
  • 4.6: [[Orthogonal Vectors in R^n.]]
  • 4.7: [General Vector Spaces.]
• 5. Linear Equations of Higher Order.
  • 5.1: Introduction: Second-Order Linear Equations.
  • 5.2: General Solutions of Linear Equations. (de-emphasize n>2)
  • 5.3: Homogeneous Equations with Constant Coefficients.
  • 5.4: Mechanical Vibrations.
  • 5.5: Nonhomogeneous Equations and Undetermined Coefficients.
    (de-emphasize most general case) [[omit variation of parameters]]
  • 5.6: Forced Oscillations and Resonance. (pick carefully from too much material here)
• 6. Eigenvalues and Eigenvectors.
  • 6.1: Introduction to Eigenvalues.
  • 6.2: Diagonalization of Matrices.
  • 6.3: [Applications Involving Powers of Matrices.]
• 7. Linear Systems of Differential Equations.
  • 7.1: First-Order Systems and Applications.
  • 7.2: Matrices and Linear Systems.
  • 7.3: The Eigenvalue Method for Linear Systems.
  • 7.4: Second-Order Systems and Mechanical Applications.
  • 7.5: Multiple Eigenvalue Solutions. (non-defective matrices only)
  • 7.6: [[Numerical Methods for Systems.]]

MAPLE

Because of the increasing number of freshmen with advanced credit, little Maple knowledge can be assumed but this is not a significant problem with the increasingly user friendly "clickable calculus" interface philosophy which sidesteps syntax and commands. Introducing MAPLE example and template worksheets associated with textbook homework problems to use MAPLE as a minimal support tool is a good idea, allowing graphing calculators or Maple to substitute for some required mechanical steps in quizzes and tests, as well as to check any hand calculations requested. Most calculators can now do row ops and row reductions.

Although the Edwards and Penney website has on-line MAPLE projects in PDF format and the corresponding MAPLE worksheets:  http://wps.prenhall.com/esm_edwards_dela_2
they are not useful in practice in our course. It is more important to have students use a limited number of MAPLE evaluation tasks on regular homework problems assigned from the textbook. In particular, the solution of any set of DEs plus initial conditions should be always available as a check for all students.

In chapter 1, DEplot for directionfields and dsolve for exact solutions (Project 1.6) should be introduced and used together with some homework problems from the text.

In chapter 3 the linear solve tutor and the LinearAlgebra package (Red...etc, BackwardsSubstitute) should be used for automated row operations and reduction and automated solution of linear systems should be covered. Technology should be emphasized for doing the row operations, since it is extremely difficult to do all the arithmetic in row reduction by hand correctly and arithmetic is not the point: the sequence of row operations is. Later in the course solving the linear system is not the main point, and the complete row reduction should be done at once with technology. Students should know how to compute determinants with technology so they can use their values to draw conclusions, after having at least one technology experience using row reduction without MultiplyRow to evaluate a determinant. Then in chapter 6 the eigenvector tutor and right click eigenvector evaluation should be introduced.

In chapter 5, solve and fsolve or right click access to them should be used for higher order (even quadratic!) polynomial roots.

In chapter 6 and 7, the DEplot command should be extended from chapter 1 to include the phase plane plots for 2-D linear systems and used to motivate and visualize eigenvectors using 2x2 matrices.

Maple specific hints:

• For stating differential equations using prime notation, the default differentiation variable x is assumed. Surrounding one or more differential equations and initial conditions by set curly braces and right-clicking on the output allows Solve DE Interactively to bring up an applet, where one can choose Solve Symbolically, then Solve
  > { x1''=x2, x2''=x1,x1(0)=1,x1'(0)=0,x2(0)=0,x2'(0)=1}
• If you want a different default differentiation variable like t without being bothered to change it, simply use explicit function notation with the desired variable:
  >  { x1''(t)=x2(t), x2''(t)=x1(t),x1(0)=1,x1'(0)=0,x2(0)=0,x2'(0)=1}
• Don't waste time using subscripted variables like x₁ with prime notation, just call it x1.
• Matrices can be entered with the Matrix palette. A superscript of -1 will produce the inverse of a square matrix, while a space " " between matrices will multiply them, without loading the LinearAlgebra or Student[LinearAlgebra] packages. To multiply a matrix by a scalar variable other than a hard number, you must use the asterisk "*" between the scalar and the matrix: 2 A but x*A, where the asterisk is then converted to a centered dot by the 2d input interpreter. Matrices can also be directly entered using < > to enclose rows or lists of rows, commas to separate entries in a Vector or separate entries vertically in a column and " | " the vertical symbol to separate entries horizontally in a row.
• Right-clicking on a matrix and selecting Standard Operations allows the determinant to be evaluated. Selecting Eigenvalues, etc allows one to get the eigenvalues and eigenvectors, or the preliminary characteristic polynomial.

Syllabus Comments by Course Coordinator

Coverage of the syllabus is a tricky problem here because combining the two topics of differential equations and linear algebra together in one semester requires cutting interesting parts of both. However, this is a terminal course so it is perhaps more important that those topics which are covered convey the enthusiasm of the instructor and accomplish student learning, even at the sacrifice of giving less attention to other parts of the syllabus (particularly those which occur at the end of the semester when time is running out). Thus the individual instructor must decide how to streamline certain parts of the syllabus in order to compensate for the extra attention given to others. Sections which are marked as optional can be mined for whatever interesting example that appeals to a given instructor, if desired, but sparingly.

One can easily overspend time in the first two chapters which is full of applications and optional material, so one must take care to pick wisely. One can streamline the first three sections of chapter 3, emphasizing the rref reduction and relying on MAPLE (optionally graphing calculators) to perform the reduction in practice. Interpretation of the rref form is more important than the distinction between Gauss and Gauss-Jordan reduction, and hand computing determinants can also be de-emphasized. [Row and column cofactor expansions should NOT be covered.] Determinants should be evaluated initially only using row reduction to triangular form without MultiplyRow operations so that a zero or nonzero value of the determinant is related to invertibility of the matrix. Later either MAPLE or graphing calculators  should be used to evaluate determinants in practice.

Emphasis can be given to section 3 in chapter 4 on the R^n vector spaces, streamlining the other 3 required sections. Section 4.6 is not very effective since the most interesting example of non-R^n vector spaces, the solution space of a linear second order DE, has not been covered yet, so the example there is unnatural. The main point that the student should understand is that solving a homogeneous linear system A x = 0 is equivalent to checking on the linear independence of the columns of A, while solving a nonhomogeneous linear system A x = b is equivalent to trying to express the vector b as a linear combination of the columns of A.

In chapter 5, one can give section 2 light treatment, omitting Wronskians, and one can lighten the undetermined coefficient section 5 by not dwelling too much on the most general case. [In practice only constant, exponential and sinusoidal driving functions are of much use.] And section 6 is really full but in practice, this is perhaps the most useful knowledge to take away from the course as far as single higher order DEs are concerned, especially the idea of resonance.

In chapter 6, it makes sense to first motivate eigenvectors with DEplot to show the eigenvector solutions of the linear DE system  x'=A x (not introduced till chapter 7) rather than just doing them first without previewing why they are needed.

In chapter 7, one can omit the subsection "simple 2-D systems" in section 1 and only expose students to the idea of reduction of order being necessary to reduce coupled damped oscillator systems to first order form. No need to worry about Wronskians in 7.2. Note that students have trouble with complex arithmetic (algebra!) which requires review, but complex eigenvectors should certainly be covered. The second order undamped multiple spring systems should be covered as the final topic, since this unites both chapter 5 and the eigenvector technique, tying together the major concepts of the course and provides a toy system with some connection to everyday intuition.