MATH 238 Differential Equations (5 credits)
Distribution Area Fulfilled Natural Sciences; Quantitative and Symbolic Reasoning; General Transfer Elective
Prerequisite MATH& 163 or equivalent with a grade of 2.0 or better (MATH 205 recommended)
Course Description
This course covers first and second order differential equations with applications to the sciences and engineering, an introduction to higher order equations, Laplace transforms, and systems of linear differential equations.
Course Content
First Order Equations:
A. Analytically solve linear and nonlinear first order equations utilizing separation of variables, integrating factors, substitutions, and exactness.
B. Graphically solve linear and nonlinear first order equations via slope fields.
C. Numerically approximate linear and nonlinear first order equations via Euler's method.
D. Determine on which intervals a given initial value problem will have a unique solution.
E. Model and solve diverse problems in the sciences and engineering via first order equations.
Second and Higher Order Equations:
F. Analytically solve second order, linear, homogeneous differential equations.
G. Utilize the Wronskian to determine the linear independence of a set of solutions.
H. Apply the methods of undetermined coefficients and variation of parameters to solve second order, linear, non-homogeneous differential equations.
I. Model and solve diverse problems in the sciences and engineering via second order equations. Higher order component
J. Utilize the Wronskian to determine the linear independence of a set of solutions.
K. Solve higher order linear, homogeneous, constant coefficient differential equations.
L. Determine the form of a specific solution to higher order linear, non-homogeneous differential equations via undetermined coefficients.
Laplace Transformations
M. State the definition of the Laplace transform of a function.
N. Compute the Laplace transform of sinusoidal, exponential, polynomial, impulse, and Dirac delta functions, as well as linear combinations of these functions and piecewise defined functions with these classes as components.
O. Apply the properties of the Laplace transform to transform a differential equation into an algebraic equation.
P. Compute the inverse Laplace transform of rational algebraic functions.
Q. Apply the methods of Laplace and inverse Laplace transforms to solve initial value problems.
R. Utilize the techniques of Laplace and inverse Laplace transforms to model and solve a diverse body of problems from the sciences and engineering.
Systems of Linear Differential Equations
S. Solve systems of linear differential equations with constant coefficients by elimination.
T. Solve systems of linear, homogeneous differential equations via eigenvalue methods, including distinct real, repeated real, and complex eigenvalues.
U. Solve systems of linear, non-homogeneous differential equations via eigenvalue methods.
V. Utilize the techniques of solving systems of equations to model and solve a diverse body of problems from the sciences and engineering.
General Content and Writing
W. Write clear, correct, and complete solutions to mathematical problems utilizing proper mathematical notation and appropriate language.
X. Compute basic examples and concepts by hand, such as, but not limited to, first and second order linear, homogeneous, constant coefficients differential equations, separable differential equations, specific solutions to undetermined coefficients and variation of parameter problems, Laplace and inverse Laplace transforms, and the eigenvalues and eigenvectors of a matrix.
Y. Solve differential equations and mathematical models utilizing an appropriate computer algebra system.
Z. Write clear, coherent, and correct mathematical proofs at a basic level, including construction of counter examples and proof by contradiction.
Student Outcomes
1. Solve first order, second order, higher order, and systems of differential equations using all the analytic, graphical, and numeric techniques described in the course content, including use of the Laplace transform.
2. Model and solve diverse problems in sciences and engineering using differential equations.
3. Write clear, correct, and complete solutions to mathematical problems and basic mathematical proofs.
Degree Outcomes
Quantitative & Symbolic Reasoning: Graduates utilize mathematical, symbolic, logical, graphical, geometric, or statistical analysis for the interpretation and solution of problems in the natural world and human society.
Critical, Creative and Reflective Thinking: Graduates will evaluate, analyze, synthesize, and generate ideas; construct informed, meaningful, and justifiable conclusions; and process feelings, beliefs, biases, strengths, and weaknesses as they relate to their thinking, decisions, and creations.
Effective Communication: Graduates will be able to exchange messages in a variety of contexts using multiple methods.
Lecture Contact Hours 50
Lab Contact Hours 0
Clinical Contact Hours 0
Total Contact Hours 50
Potential Methods
A. TEST - multiple choice, true/false, computational, matching, or short written answers.
B. WRITING ASSIGNMENT - journal, outline, narrative explanation, essay, notebook, portfolio, or projects.
C. GROUPWORK - written group project, group written solution to problems, group assignment, group presentation, group oral solution to problems, or group discussion.
D. INDIVIDUAL WORK - individual presentation, individual written solution to problems, or individual oral solution to problems.
E. INFORMAL ASSESSMENT - self-evaluation, peer evaluation, or teacher observation.
F. CRITICAL THINKING RUBRIC.
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