Description:

The focus of the course will be the study and mathematical modeling of engineering systems, both mechanical and electrical. Solution techniques, both analytic and numeric, for a single ordinary differential equation and for systems of first-order ordinary differential equations are used. Also, Laplace transforms and their applications are used as they apply to engineering systems. Linear algebraic systems of equations and the concepts of vector spaces, basis, dimension and subspaces are encountered as well. 5 hrs. lecture/wk.

Supplies: Refer to the instructor’s course syllabus for details about any supplies that may be required.

Textbook(s): For information see - http://bookstore.jccc.net

Course Fees: NONE

Course Objectives:

Upon successful completion of this course the student should be able to:

1. Calculate solutions to first-order ordinary-differential equations;
2. Apply numerical methods in solving first-order initial-value problems;
3. Calculate solutions to second- and higher-order ordinary-differential equations;
4. Calculate solutions to systems of first-order ordinary-differential equations;
5. Apply numerical methods in solving systems of first-order initial-value problems;
6. Utilize Laplace transforms in solving ordinary differential equations;
7. Analyze engineering systems in the context of Laplace transform calculus;
8. Solve linear algebraic systems utilizing Gaussian elimination;
9. Utilize the concepts of vector spaces, basis, dimension, & subspace in solving engineering systems;
10. Utilize the above objectives in applied modeling activities, particularly for electrical and mechanical systems.

Content Outline & Competencies:

I. First-Order Ordinary Differential Equations 
   A. Define ordinary versus partial differential equations;
   B. Define the degree of a differential equation;
   C. Define linear versus nonlinear differential equations;
   D. Define a solution to a differential equation;
   E. Utilize direction fields in solving differential equations;
   F. Introduce the linear existence and uniqueness theorem;
   G. Calculate solutions to linear first-order ordinary differential
equations;
   H. Introduce the nonlinear existence and uniqueness theorem(s);
   I. Calculate solutions to separable first-order ordinary differential
equations;
   J. Define the criteria for exactness;
   K. Calculate solutions to exact first-order ordinary differential
equations;
   L. Calculate solutions to homogeneous first-order ordinary differential
equations;
   M. Utilize first-order ordinary differential equations in modeling
activities, particularly models of mixing;
   N. Define the Euler method;
   O. Define improvements on the Euler method;
   P. Define the Runge-Kutta methods;
   Q. Introduce Predictor-corrector methods;
   R. Utilize numerical methods in solving first-order initial value
problems. 

II. Higher-Order Ordinary Differential Equations 
   A. Calculate solutions to homogeneous higher-order ordinary
differential equation with constant coefficients;
   B. Define fundamental solutions of linear homogeneous equations;
   C. Define linear dependence and independence;
   D. Utilize the Wronskian to determine the linear independence of
solutions of equations;
   E. Utilize the technique of reduction of order in solving linear
second-order ordinary differential equations;
   F. Calculate solutions given distinct real roots of the characteristic
equation;
   G. Calculate solutions given complex roots of the characteristic
equation;
   H. Calculate solutions given repeated real roots of the characteristic
equation;
   I. Define Euler equations;
   J. Solve Euler equations;
   K. Utilize the undetermined coefficients method in solving
nonhomogeneous equations;
   L. Utilize the variation of parameters method in solving nonhomogeneous
equations;
   M. Utilize higher-order ordinary differential equations in mechanical
system modeling activities;
   N. Utilize higher-order ordinary differential equations in electrical
circuit modeling activities;
   O. Utilize higher-order ordinary differential equations in applied
modeling activities. 

III. Laplace Transforms 
   A. Define the Laplace transform;
   B. Utilize Laplace transforms in solving initial-value problems;
   C. Define step functions and their Laplace transforms;
   D. Utilize theorems involving translation, derivatives, & integrals;
   E. Utilize the convolution theorem as it relates to products of
transforms;
   F. Work problems involving discontinuous forcing functions;
   G. Utilize the concepts of impulse and delta functions and their
Laplace transforms;
   H. Utilize Laplace transforms in analyzing systems;
   I. Utilize Laplace transforms in applied modeling activities. 

IV. Linear Algebra 
   A. Perform elimination using pivots, basic and free variables for a
given system of equations;
   B. Define the determinant of an n-by-n matrix;
   C. Utilize the determinant of an n-by-n matrix;
   D. Represent a system of linear algebraic equations using vectors;
   E. Define vector spaces and subspaces;
   F. Define linear independence and dependence;
   G. Define a basis for a vector space or subspace;
   H. Define the dimension for a vector space;
   I. Determine the null space and solution space for a given system of
equations;
   J. Calculate the eigenvalues and associated eigenvectors for a given
matrix. 

V. Systems of Linear Ordinary Differential Equations 
   A. Convert higher-order differential equations into a system of
first-order equations;
   B. Utilize the eigenvalue-eigenvector method in solving systems of
homogeneous linear differential equations with constant coefficients;
   C. Calculate solutions to a homogeneous linear system given distinct
real eigenvalues;
   D. Calculate solutions to a homogeneous linear system given
complex-conjugate eigenvalues;
   E. Calculate solutions to a homogeneous linear system given repeated
real eigenvalues;
   F. Construct the general solution of nonhomogeneous linear systems;
   G. Introduce stability considerations;
   H. Utilize the phase plane in solving systems;
   I. Analyze the solutions of almost linear systems;
   J. Utilize systems of linear differential equations in applied modeling
activities.

Methods of Evaluation of Competencies:

Evaluation of student mastery of course competencies will be accomplished using the following methods:

Unit Exams, Unit Papers and/or Unit Projects   40% - 80%
Homework, Quizzes and/or Small Projects         0% - 50%
Final Exam**                                   10% - 40%

**The final exam must count at least as much as any unit exam, unit paper
or unit project. In any course where unit exams are not proctored, the
instructor may require that the student score at least a 70% on the final
exam to earn a ‘C’ for the course. At the instructor's discretion, the
grade on all or any part of the final exam may replace any lower test
score.

Caveats:

1. The majority of mathematics courses are sequential. Students must earn a grade of C or higher in a prerequisite mathematics course to progress to its subsequent mathematics course.
2. In accordance with the assertion made on your billing statement, during the first two weeks of the semester, if a student not to have successfully fulfilled the prerequisite(s) for this course, the student will be dropped from the course. He/she will be allowed to enroll in the appropriate lower level course on a space-available basis with an even exchange of tuition. After the first two weeks, students who have not met the prerequisite(s) will be dropped from the course with no refund of tuition.

Disabilities:

If you are a student with a disability, and if you will be requesting accommodations, it is your responsibility to contact Access Services. Access Services will recommend any appropriate accommodations to your professor and his/her director. The professor and director will identify for you which accommodations will be arranged.

JCCC provides a range of services to allow persons with disabilities to participate in educational programs and activities. If you desire support services, contact the office of Access Services for Students With Disabilities (913) 469-8500, ext. 3521 or TDD (913) 469-3885. The Access Services office is located in the Success Center on the second floor of the Student Center.