El Camino College  Division of
Mathematical Sciences
Math
270
Differential Equations with Linear Algebra
5 units;
5 hours lecture
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 firstorder ordinary differential
equations, systems of linear equations, matrices, determinants, vector
spaces, linear transformations, linear secondorder 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:
 Course objectives (list
the major
objectives stated as student outcomes in behaviorally measurable
terms.)
 Solve various
types of ordinary differential equations: separable; exact; firstorder
linear; two special types of secondorder; linear, higherorder
differential equations with constant coefficients; CauchyEuler
equations, and non homogeneous equations.
 Solve differential
equations using the following numerical methods, including: Euler
method, Taylor series methods, and RungeKutta methods.
 Find power series
solutions to differential equations.
 Perform operations on
matrices and prove theorems involving matrices.
 Prove theorems about
determinants and solve problems involving determinants.
 Solve linear systems of
equations, both dependent and independent.
 Determine whether a
given set constitutes a vector space or a subspace of a known vector
space.
 Determine whether a
given set of vectors or functions is independent.
 Determine whether a set
of vectors spans a given vector space.
 For some common vector
spaces find a basis and the dimension, and prove the result.
 Use the GramSchmidt
procedure to find an orthonormal basis for a given subspace.
 Determine whether or
not a given operator is a linear transformation.
 Carry out a variety of
proofs and problems involving the kernel, range, composition and
inverse of linear transformations.
 Carry out a variety of
proofs and problems involving the kernel, range, composition and
inverse of linear transformations.
 Work with differential
operator notation.
 Find eigenvalues and
eigenvectors of a matrix.
 Solve systems of
firstorder linear differential equations using eigenvectors.
 Find the Laplace and
inverse transformations of various functions using the definition,
tables and shifting theorems.
 Solve differential
equations using Laplace transforms.
 Use a computer algebra
system to solve problems in differential equations and linear algebra;
and solve application problems.
 Methods of Evaluation 
Associate
Degree Credit Course
 Substantial writing
assignments are
inappropriate for this degree applicable course because:
 The course is primarily
computational
in nature.
 The course primarily
involves skill
demonstrations or problem solving.
 Computational or
noncomputational
problemsolving demonstrations, including:
 Exam
 Quizzes
 Homework Problems
Return
to the
top of the page.
Outline of Subject
Matter
Approximate
Time

Major
Topic

10 hours

I. Introduction to Differential Equations
 Introduction
 Separable Equations
 Homogeneous Equations
 Exact Equations
 FirstOrder Equations
 Orthogonal Trajectories
 Two Special Types of
SecondOrder Equations

15
hours

II. Matrices and
Determinants
 Systems of Linear
Equations
 Homogeneous Systems
 Matrices and Vectors
 Matrix Multiplication
 Inner Product and Length
 Some Special Matrices
 Definition of Determinant
 Properties of Determinants
 Cofactors
 Cramer's Rule
 The Inverse of a Matrix
 Proofs involving matrix
algebra, e.g., A(B + C) = AB + AC
 Proofs involving
Determinants

15 hours

III. Vector Spaces
and Linear Transformations
 Vector Spaces and
Subspaces
 Linear Independence and
Spanning
 Wronskians
 Dimension and Bases
 Orthogonal Bases
 Linear Transformations:
kernel, range, composition, inverse
 Properties of Linear
Transformations
 Proofs involving vector
spaces and linear transformations
 Differential Operators

15
hours

IV. Linear
Differential Equations
 Introduction to Linear
Differential Equations
 Polynomial Operators
 Complex Solutions
 Equations with Constant
Coefficients
 CauchyEuler Equations
 Nonhomogeneous Equations
 The Method of Undetermined
Coefficients
 Variations of Parameters

5 hours

V. Eigenvalues,
Eigenvectors and Systems of Differential Equations
 Eigenvalues and
Eigenvectors
 Solutions of Systems
of Differential Equations by eigenvectors

10 hours 
VI. Power Series
Solutions
Differential Equations and Numerical Methods
 Construction of Taylor
Series
 Series Solutions of
Differential Equations with Ordinary Points
 Series Solutions of
Differential Equations with Singular Points
 The Euler Method,
Taylor Series Methods, RungeKutta Methods of Finding Numerical
Solutions of Differential Equations

10 hours 
VII. Laplace
Transforms
 The Laplace Transform
 Functions of
Exponential Order
 Properties of Laplace
Transforms
 Inverse Transforms
 Applications to
Differential Equations
 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

Return
to the
top of the page.
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.
 Calculate partial derivatives and double
integrals.
Source of
information: Course Outline of Record dated November, 1999
Last Updated On: 4/20/06