## M427K Syllabus

ADVANCED CALCULUS FOR APPLICATIONS I

**Prerequisite and degree relevance:** The prerequisite is one of M408D, M308L, M408L, M308S, M408S or M408M with a grade of at least C-.

**Course description:** M427K is a basic course in ordinary and partial differential equations, with Fourier series. It should be taken before most other upper division, applied mathematics courses. The course meets three times a week for lecture and twice more for problem sessions. Geared to the audience primarily consisting of engineering and science students, the course aims to teach the basic techniques for solving differential equations which arise in applications. The approach is problem-oriented and not particularly theoretical. Most of the time is devoted to first and second order ordinary differential equations with an introduction to Fourier series and partial differential equations at the end. Depending on the instructor, some time may be spent on applications, Laplace transformations, or numerical methods. Five sessions a week for one semester.

Note that some Engineering courses assume students have seen Laplace Transforms in M427K.

**Text: Boyce and DiPrima: Elementary Differential Equations and Boundary Value Problems. 10th Edition** The text is required for most sections; honors classes, computer supplement sections, or innovative sections may use other texts.

**Required Topics**

It will be impossible to cover everything here adequately. The core material which must be covered is selected sections from Chapters 1, 2, 3, 5, 10. Chapter 7 is so important that it ought to be covered, but be aware that most students have not already had matrix methods, and you will likely find yourself covering the 2 by 2 case. You might then do stability, etc. Numerical methods are becoming increasingly important, and covering this topic here is a good lead in to the departments new computational science degree. Again, some engineering courses need their students to have seen some Laplace transforms. This will leave time for other topics, and you may choose to equations, applications. Whichever approach you take, you will have to carefully plan your sections and time to be spent on them.

**Resources**

If you are new to this course, you might talk to the senior faculty who teach this course regularly: Profs. Arbogast, Beckner, Bichteler, Gamba, Koch, Tsai, Uhlenbeck and others.

- Chapter I Introduction (2 - 3 weeks for Chapters 1 and 2)
- 1.1Some Basic Mathematical Models; Direction Fields
- 1.2 Solutions of Some Differential Equations 1.3 Classification of Differential Equations
- 1.4 Historical Remarks

- Chapter 2 First Order Differential Equations(2 - 3 weeks for Chapters 1 and 2)
- 2.1 Linear Equations with Variable Coefficients
- 2.2 Separable Equations
- 2.3 Modeling with First Order Equations (optional)
- 2.4 Differences Between Linear and Nonlinear Equations
- 2.5 Autonomous Equations and Population Dynamics (optional)
- 2.6 Exact Equations and Integrating Factors
- 2.7 Numerical Approximations: Euler's Method (optional unless you do Ch 8)
- 2.8 The Existence and Uniqueness Theorem
- 2.9 First Order Difference Equations 115 (optional unless you do stability)

- Chapter 3 Second Order Linear Equations (2 - 3 weeks)
- 3.1 Homogeneous Equations with Constant Coefficients
- 3.2 Fundamental Solutions of Linear Homogeneous Equations
- 3.3 Complex Roots of the Characteristic Equation
- 3.4 Repeated Roots; Reduction of Order
- 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
- 3.6 Variation of Parameters (optional)
- 3.7 Mechanical and Electrical Vibrations (optional)
- 3.8 Forced Vibrations (optional)

- Chapter 4 Higher Order Linear Equations (cover quickly)
- 4.1 General Theory of nth Order Linear Equations
- 4.2 Homogeneous Equations with Constant Coefficients
- 4.3 The Method of Undertermined Coefficients (optional)
- 4.4 The Method of Variation of Parameters

- Chapter 5 Series Solutions of Second Order Linear Equations (2 weeks)
- 5.1 Review of Power Series (optional)
- 5.2 Series Solutions near an Ordinary Point, Part I
- 5.3 Series Solutions near an Ordinary Point, Part II
- 5.4 Euler Equations, Regular Singular Points
- 5.5 Series Solutions near a Regular Singular Point, Part I
- 5.6 Series Solutions near a Regular Singular Point, Part II

- Chapter 6 The Laplace Transform (1 week: Important for some Engineers))
- 6.1 Definition of the Laplace Transform
- 6.2 Solution of Initial Value Problems
- 6.3 Step Functions
- 6.4 Differential Equations with Discontinuous Forcing Functions
- 6.5 Impulse Functions
- 6.6 The Convolution Integral

- Chapter 7 Systems of First Order Linear Equations (1 – 2 weeks)

- Chapter 8 Numerical Methods (1 week if covered) (optional)

- Chapter 9 Nonlinear Differential Equations and Stability

- Chapter 10 Partial Differential Equations and Fourier Series (3 weeks)

- 10.1 Two Point Boundary Value Problems
- 10.2 Fourier Series
- 10.3 The Fourier Convergence Theorem
- 10.4 Even and Odd Functions
- 10.5 Separation of Variables; Heat Conduction in a Rod
- 10.6 Other Heat Conduction Problems (optional)
- 10.7 The Wave Equation; Vibrations of an Elastic String
- 10.8 Laplace's Equation (optional)
- Appendix A Derivation of the Heat Equation (optional)
- Appendix B Derivation of the Wave Equation (optional)