Math 245 does not have a common final exam.

REQUIRED TEXTBOOK

Brannan & Boyce, Differential Equations (3rd ed.)

SECTION COVERAGE

The following table lists the minimum set of topics to be included in this course.  The number of lectures listed for each chapter is only a suggestion and will vary across instructors and semesters.

There are typically 42-43 lecture days in a semester, so lecture periods are available for exams.

 Sections Topics Lectures 1.1 – 1.3 Introduction and Qualitative Methods 3 2.1, 2.2, 2.4, 2.6 First Order Differential Equations 4 3.1 – 3.5 Systems of Two First Order Equations 8 4.1 – 4.7 Second Order Linear Equations 7 5.1 – 5.8 The Laplace Transform 9 A.1 – A.4 Linear Algebra 6 6.2 – 6.5 Systems of First Order Linear Equations 4 Total 41

MEASURABLE OBJECTIVES

By the end of this course, students should be able to:

1. Define and explain fundamental concepts in linear algebra, such as vectors, matrices, scalars, and vector spaces.
2. Perform basic operations on vectors and matrices, including addition, subtraction, scalar multiplication, and matrix multiplication.
3. Analyze the properties of vector spaces, including basis, dimension, linear independence, and span.
4. Solve systems of linear equations using various methods, including Gaussian elimination and matrix inversion.
5. Calculate the determinant of a matrix and understand how row operations affect the determinant.
6. Determine the invertibility of a matrix and calculate matrix inverses using Gauss-Jordan technique.
7. Apply techniques for finding eigenvalues and eigenvectors of square matrices and understand their significance in various applications.
8. Understand the basic terminology and concepts related to differential equations, including order, degree, linearity, and initial/boundary value problems.
9. Solve first-order ordinary differential equations (ODEs) both analytically and numerically using separation of variables, integrating factors, and numerical methods like Euler’s method.
10. Apply the concept of direction fields to visualize and analyze solutions of first-order ODEs.
11. Solve higher-order linear ODEs with constant coefficients using characteristic equations and find the general solution.
12. Use the method of undetermined coefficients and variation of parameters to solve non-homogeneous ODEs.
13. Understand and solve systems of first-order linear ODEs using matrix methods and eigenvalues/eigenvectors.
14. Analyze and solve applications of ODEs in various fields, such as physics, engineering, biology, and economics.
15. Understand the concept of Laplace transforms and apply them to solve initial value problems and piecewise-defined functions.
16. Use convolution to solve linear constant coefficient differential equations.
17. Solve second-order ODEs with variable coefficients using power series methods (Frobenius method).
18. Explore the concept of series solutions for ordinary differential equations and apply the method of Frobenius to find solutions near singular points.
19. Analyze and solve higher-order linear ODEs using the concept of recurrence relations and generating functions.
20. Understand the concept of stability and apply it to analyze solutions of first-order ODEs and systems of ODEs.
21. Analyze phase portraits and stability of equilibrium points for systems of first-order ODEs.
22. Introduce the basic concepts of partial differential equations (PDEs) and their classification.
23. Understand and solve simple first-order linear PDEs, such as the heat equation and the wave equation.
24. Apply separation of variables to solve simple partial differential equations with boundary conditions.
25. Use Fourier series to solve boundary value problems involving partial differential equations.