# Applied Differential Equations and Modeling

### Syllabus

Summary:
This course offers an introduction to ordinary differential equations and their applications. Mathematical modeling of continuous-time dynamics has its origins in classical mechanics but is now prevalent in all areas of physical and life sciences. Attempting to solve such problems often leads to a differential equation. Consequently, a variety of analytical and numerical methods have been developed to deal with various classes of equations and initial value problems, the most important of which is the class of linear equations. Other methods (such as Laplace transform) for solving many differential equations of special form will also be discussed.

Contact Information:
 Instructor: Marcel Oliver Email: m.oliver@jacobs-university.de Phone: 200-3212 Office hours: Mo 11:30, We 10:00 in Research I, 107 TA: TBA

Time and Place:
 Lectures: Mo/We 11:15-12:30 in East Hall 2 Discussions: Th, 13:15 in East Hall 2

• J. Brannan and W. Boyce, Differential equations. An introduction to modern methods and applications, 3rd edition, 3rd edition, Wiley, 2015.

• The final grade will be determined as follows:

 Homework: 30% Midterm Exam: 30% Final Exam: 40%

• The scores will be averaged as percentage grades with the given weights.
• Homework will be given every week, it is due in class the following Monday.
• I reserve the right to substitute in-class Quizzes for part of the homework score.

### Class Schedule (subject to change!)

 Feb. 4., 2019: Introduction, Overview, First example: Newton's law of cooling (Brannan/Boyce, Section 1.1) Feb. 6., 2019: Integrating factors and separation of variables (Brannan/Boyce, Sections 1.2 and 2.1) Feb. 12., 2019: Detailed discussion of long-time behavior, modeling examples (Brannan/Boyce, Section 2.2) Feb. 13., 2019: Differerence between linear and nonlinear equations; existence, uniqueness; examples of non-uniqueness and blow-up (Brannan/Boyce, Section 2.3) Feb. 19., 2019: Population dynamics and logistic growth; optimal harvesting (Brannan/Boyce, Section 2.4) Feb. 20., 2019: Exact equations (one example), elementary discussion of numerical methods (Brannan/Boyce, selected topics from Sections 2.5-2.7) Feb. 26., 2019: Exact equations and integrating factors (Brannan/Boyce, Section 2.5) Feb. 27., 2019: Numerical methods (Brannan/Boyce, Section 2.6) Mar. 5., 2019: Trapezoidal rule and Runge-Kutta methods (Brannan/Boyce, Section 2.7) Mar. 6., 2019: Crash course in Linear Algebra I (Brannan/Boyce, Section 3.1) Mar. 12., 2019: Crash course in Linear Algebra I Mar. 13., 2019: Systems of two first-order linear equations (Brannan/Boyce, Section 3.2 and 3.3) Mar. 19., 2019: Systems of two first-order linear equations, complex and repeated eigenvalues (Brannan/Boyce, Section 3.4 and 3.5) Mar. 20., 2019: Midterm review Mar. 26., 2019: Midterm Exam Mar. 27., 2019: Second order linear equations I (selected topics from Brannan/Boyce, Chapter 4) Apr. 2., 2019: Second order linear equations I (selected topics from Brannan/Boyce, Chapter 4) Apr. 3., 2019: The Laplace transform (Brannan/Boyce, Section 5.1 and 5.2 Apr. 9., 2019: The inverse Laplace transform (Brannan/Boyce, Section 5.3) Apr. 10., 2019: Solving differential equations via the Laplace transform (Brannan/Boyce, Section 5.4 Apr. 23., 2019: TBA Apr. 24., 2019: Discontinuous and periodic functions, applications to forcing (Brannan/Boyce, Sections 5.5-5.7 Apr. 30., 2019: Convolution Integrals, linear systems, and feedback control (selected topics from Brannan/Boyce, Sections 5.8 and 5.9) May 7., 2019: Linear stability (Brannan/Boyce, Section 7.1 May 8, 2019: Almost linear systems (Brannan/Boyce, Section 7.2 May 14, 2019: Predator-prey systems (Brannan/Boyce, Section 7.3 and 7.4) May 15, 2019: Final exam review TBA: Final Exam, Room TBA