{"id":1450,"date":"2023-02-15T12:04:50","date_gmt":"2023-02-15T20:04:50","guid":{"rendered":"https:\/\/dornsife.usc.edu\/mathematics\/?page_id=1450"},"modified":"2023-08-05T15:07:59","modified_gmt":"2023-08-05T22:07:59","slug":"math-245-mathematics-of-physics-and-engineering-1","status":"publish","type":"page","link":"https:\/\/dornsife.usc.edu\/mathematics\/math-245-mathematics-of-physics-and-engineering-1\/","title":{"rendered":"MATH 245 – Mathematics of Physics and Engineering I"},"content":{"rendered":"\n\n \n \n\n\n\n\n\n\n\n
\n\n \n \n
\n\n \n

Math 245 does not have a common final exam.<\/span><\/p>\n

 <\/p>\n

REQUIRED TEXTBOOK<\/strong><\/p>\n

Brannan & Boyce, Differential Equations (3rd ed.)<\/a><\/p>\n

 <\/p>\n

SECTION COVERAGE<\/strong><\/p>\n

The following table lists the minimum<\/em> set of topics to be included in this course.\u00a0 The number of lectures listed for each chapter is only a suggestion and will vary across instructors and semesters.<\/p>\n

There are typically 42-43 lecture days in a semester, so lecture periods are available for exams.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n
Sections<\/strong><\/td>\nTopics<\/strong><\/td>\nLectures<\/strong><\/td>\n<\/tr>\n
1.1 – 1.3<\/td>\nIntroduction and Qualitative Methods<\/td>\n3<\/td>\n<\/tr>\n
2.1, 2.2, 2.4, 2.6<\/td>\nFirst Order Differential Equations<\/td>\n4<\/td>\n<\/tr>\n
3.1 – 3.5<\/td>\nSystems of Two First Order Equations<\/td>\n8<\/td>\n<\/tr>\n
4.1 – 4.7<\/td>\nSecond Order Linear Equations<\/td>\n7<\/td>\n<\/tr>\n
5.1 – 5.8<\/td>\nThe Laplace Transform<\/td>\n9<\/td>\n<\/tr>\n
A.1 – A.4<\/td>\nLinear Algebra<\/td>\n6<\/td>\n<\/tr>\n
6.2 – 6.5<\/td>\nSystems of First Order Linear Equations<\/td>\n4<\/td>\n<\/tr>\n
<\/td>\nTotal<\/td>\n41<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

MEASURABLE OBJECTIVES<\/strong><\/p>\n

By the end of this course, students should be able to:<\/p>\n

 <\/p>\n

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

     <\/p>\n

     <\/p>\n

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