7a. Prerequisite(s): (Course and/or other preparation/experience that is REQUIRED to be completed previous to enrollment in this course.)

Completion of Math. 31 with a grade of "C" or better

7b. Co-requisite(s):  (Courses and/or other preparation that is REQUIRED to be taken concurrently with this course.)

7c. Advisory: (Minimum preparation RECOMMENDED in order to be successful in this course.  Also known as “Course Advisory”.)

Math. 32 strongly recommended

9b. Catalog Description:

First and second order ordinary differential equations, linear differential equations, numerical methods and series solutions, Laplace transforms, modeling and stability theory, systems of linear differential equations, matrices, determinants, vector spaces, linear transformations, orthogonality, eigenvalues and eigenvectors.

10. Student Performance Outcomes: (Outcomes for all credit courses must indicate that students will learn critical thinking and will be able to apply concepts at college level.  Outcomes must be related to items listed in Section 11.)

1. solve first order differential equations analytically, numerically, and graphically;
2. solve higher order differential equations;
3. construct a basis for the solution space of a differential equation;
4. apply Green's theorem to solve a differential equation;
5. perform basic operations on matrices;
6. use an augmented matrix and Gaussian elimination to solve a corresponding system of linear equations;
7. apply the inverse matrix method to solve a system of linear equations;
8. apply Cramer's rule to solve a system of linear equations;
9. verify that the axioms of a vector space, subspace, and inner product are satisfied for a variety of sets including: n-dimensional space, polynomials, matrices, continuous and differentiable functions;
10. apply the definition, the wronskian, and the determinant to determine the independence/dependence of vectors in a vector space;
11. construct the nullspace from a given matrix;
12. apply the Rank-Nullity theorem to determine the dimension of a vector space;
13. apply the Gram-Schmidt procedure to generate a set of orthogonal and orthonormal vectors that span a given space;
14. verify that a transformation is linear;
15. construct the kernel and range of a linear transformation;
16. apply eigenvalues, diagonalization and variation of parameters to solve a system of differential equations;
17. construct a matrix exponential function for a system of differential equations;
18. examine the phase plane for generating a qualitative representation of the solution to a system of nonlinear differential equations;
19. use Laplace transforms to determine the solutions to a differential equation with initial value conditions;
20. solve differential equations with forcing functions involving the unit step function and forcing functions involving the Dirac delta function;
21. apply the convolution integral to solve appropriate differential equations;
22. assess the need for the appropriate shifting theorems and apply when appropriate to solve a differential equation;
23. solve differential equations using power series methods including Frobenius solutions;
24. examine Legendre and Bessel differential equations and their solutions.

11. Course Content Outline: (Provides a comprehensive, sequential outline of the course content, including all major subject matter and the specific body of knowledge covered.)

I. First Order Differential Equations
A. Slope Fields and Isoclines
B. Separation of Variables
C. Integrating Factors
D. Bernoulli Differential Equations
E. Homogeneous First-Order Differential Equations
F. Exact Differential Equations
G. Applications to First-Order Differential Equations
H. Numerical Techniques
II. Elements of Linear Algebra
III. Linear Transformations and Linear Differential Operators
IV. Higher-Order Linear Differential Equations
A. Phase Plane
B. Homogeneous Constant Coefficient Differential Equations
C. Method of Undetermined Coefficients
D. Variation of Parameters
E. Applications of Higher-Order Differential Equations
V. Laplace Transformations
A. Inverse Laplace Transformations
B. Shifting Theroems
C. Unit Step Function
D. Dirac Delta Function
E. Convolution Integral
VI. Series Solutions
VII. Matrices and Systems of Linear Equations
VIII. Systems of Linear Differential Equations

12. Typical Assignments: (List types of assignments, including library assignments.)

a. Reading Assignments: (Submit at least 2 examples)

1. Read the text content regarding the axioms of a vector space prior to class discussion on the subtle nature of these axioms.
2. Review in a standard calculus text power series representations for functions.

b. Writing, Problem Solving or Performance: (Submit at least 2 examples)

1. Solve the differential equation D(y)+p(x)y=q(x) where p(x)=exp(x) and q(x)=exp(-x)
2. Prove that P3 is a vector space by verifying that the set P3 satisfies each of the axioms for a vector space.

c. Other (Terms projects, research papers, portfolios, etc.)

SECTION D

SECTION E

SECTION F

SECTION G