El Camino College - Division of Mathematical Sciences

Math 270
Differential Equations with Linear Algebra
5 units; 5 hours lecture

Catalog Description Course Objectives and Methods of Evaluation
Outline of Subject Matter Planned Instructional Activities

Grading Method: Letter

Associate Degree Credit --- Transfers to CSU and Transfers to UC

Prerequisite: Mathematics 220 with a minimum grade of C.

Catalog Description:
This course consists of a study of first-order ordinary differential equations, systems of linear equations, matrices, determinants, vector spaces, linear transformations, linear second-order ordinary differential equations, power series solutions, numerical methods, Laplace transforms, eigenvalues, eigenvectors, systems of linear differential equations and applications.
Note: Mathematics 270 was formerly numbered Mathematics 6B.

Course Objectives and Methods of Evaluation:

  1. Course objectives (list the major objectives stated as student outcomes in behaviorally measurable terms.)
    1. Solve various types of ordinary differential equations: separable; exact; first-order linear; two special types of second-order; linear, higher-order differential equations with constant coefficients; Cauchy-Euler equations, and non homogeneous equations.
    2. Solve differential equations using the following numerical methods, including: Euler method, Taylor series methods, and Runge-Kutta methods.
    3. Find power series solutions to differential equations.
    4. Perform operations on matrices and prove theorems involving matrices.
    5. Prove theorems about determinants and solve problems involving determinants.
    6. Solve linear systems of equations, both dependent and independent.
    7. Determine whether a given set constitutes a vector space or a subspace of a known vector space.
    8. Determine whether a given set of vectors or functions is independent.
    9. Determine whether a set of vectors spans a given vector space.
    10. For some common vector spaces find a basis and the dimension, and prove the result.
    11. Use the Gram-Schmidt procedure to find an orthonormal basis for a given subspace.
    12. Determine whether or not a given operator is a linear transformation.
    13. Carry out a variety of proofs and problems involving the kernel, range, composition and inverse of linear transformations.
    14. Carry out a variety of proofs and problems involving the kernel, range, composition and inverse of linear transformations.
    15. Work with differential operator notation.
    16. Find eigenvalues and eigenvectors of a matrix.
    17. Solve systems of first-order linear differential equations using eigenvectors.
    18. Find the Laplace and inverse transformations of various functions using the definition, tables and shifting theorems.
    19. Solve differential equations using Laplace transforms.
    20. Use a computer algebra system to solve problems in differential equations and linear algebra; and solve application problems.
  1. Methods of Evaluation - Associate Degree Credit Course
    1. Substantial writing assignments are inappropriate for this degree applicable course because:
      1. The course is primarily computational in nature.
      2. The course primarily involves skill demonstrations or problem solving.
    2. Computational or non-computational problem-solving demonstrations, including:
      1. Exam
      2. Quizzes
      3. Homework Problems

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Outline of Subject Matter
 

Approximate Time

Major Topic

10 hours

I. Introduction to Differential Equations

  1. Introduction
  2. Separable Equations
  3. Homogeneous Equations
  4. Exact Equations
  5. First-Order Equations
  6. Orthogonal Trajectories
  7. Two Special Types of Second-Order Equations

15 hours

II. Matrices and Determinants

  1. Systems of Linear Equations
  2. Homogeneous Systems
  3. Matrices and Vectors
  4. Matrix Multiplication
  5. Inner Product and Length
  6. Some Special Matrices
  7. Definition of Determinant
  8. Properties of Determinants
  9. Cofactors
  10. Cramer's Rule
  11. The Inverse of a Matrix
  12. Proofs involving matrix algebra, e.g., A(B + C) = AB + AC
  13. Proofs involving Determinants

15 hours

III. Vector Spaces and Linear Transformations

  1. Vector Spaces and Subspaces
  2. Linear Independence and Spanning
  3. Wronskians
  4. Dimension and Bases
  5. Orthogonal Bases
  6. Linear Transformations: kernel, range, composition, inverse
  7. Properties of Linear Transformations
  8. Proofs involving vector spaces and linear transformations
  9. Differential Operators

15 hours

IV.  Linear Differential Equations

  1. Introduction to Linear Differential Equations
  2. Polynomial Operators
  3. Complex Solutions
  4. Equations with Constant Coefficients
  5. Cauchy-Euler Equations
  6. Nonhomogeneous Equations
  7. The Method of Undetermined Coefficients
  8. Variations of Parameters

5 hours

V. Eigenvalues, Eigenvectors and Systems of Differential Equations

  1. Eigenvalues and Eigenvectors
  2. Solutions of  Systems of Differential Equations by eigenvectors
10 hours

VI. Power Series Solutions Differential Equations and Numerical Methods

  1. Construction of Taylor Series
  2. Series Solutions of Differential Equations with Ordinary Points
  3. Series Solutions of Differential Equations with Singular Points
  4. The Euler Method, Taylor Series Methods, Runge-Kutta Methods of Finding Numerical Solutions of Differential Equations
10 hours

VII. Laplace Transforms

  1. The Laplace Transform
  2. Functions of Exponential Order
  3. Properties of Laplace Transforms
  4. Inverse Transforms
  5. Applications to Differential Equations
  6. Functions with Discontinuities
5 hours

VIII. Problem Solving in Differential Equations and Linear Algebra Using a Computer Algebra System and Applications to Engineering and Physics.

  5 hours

Examinations

Total:

90 Hours

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Planned Instructional Activities:

Lecture, discussion, student presentation group work, and use of a computer algebra system.

Entrance Skills and Knowledge:

List the required skills and/or knowledge without which a student would be highly unlikely to receive a grade of A, B, C, or Credit (or for Health and Safety, would endanger self or others) in the Target Course.

  1. Calculate partial derivatives and double integrals.
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Source of information: Course Outline of Record dated November, 1999


 Last Updated On: 4/20/06