Differential equations are the most powerful and most pervasive mathematical tool in the sciences. Any time a law is expressed in the form “what happens in the next moment”, we have a differential equation; and determining the long-term behavior is the domain of differential equations. Planets, stars, fluids, electric circuits, predator and prey populations, epidemics: almost any system whose components interact continuously over time is modeled by a differential equation. Differential equations are fundamental in pure mathematics as well. The main emphasis of this course is on the classical theory of ordinary differential equations, as represented in the classic text of Tenenbaum and Pollard. However, there will be more emphasis than usual on actually recognizing when a situation may be modeled by a differential equation, and on setting up the differentials, in addition to finding the solutions. We will also be covering some asymptotic analysis, as in the text of Bender and Orszag. The depth of coverage may be adjusted depending on each student’s calculus background. We will cover exact solutions to linear differential equations in one variable, and asymptotic solutions for non-linear equations. More advanced qualitative analysis of non-linear equations will be covered in MAT 4108 *Differential Equations and Non-Linear Systems*, for which this class would be a good preparation.

Note that in our non-standard calculus sequence, we do not cover some standard computational techniques of calculus in the introductory course MAT 4288 *Calculus: A Classical Approach*. Those techniques are covered in this class instead. Therefore, this class is a good choice for students who took *Calculus: A Classical Approach* and want to go further with calculus.

**Learning Outcomes:**

- Develop and practice computational techniques of calculus, including derivative rules, substitution, integration by parts, and partial fractions

- Recognize when a situation may be modeled by a differential equation; find differential equation models, and evaluate their plausibility

- Carry out all the standard exact solutions of ordinary differential equations (at the level of Tenenbaum and Pollard), and do rudimentary asymptotic analysis when exact solutions are not available (at the level of introductory parts of Bender and Orszag)

- Be familiar with some of the standard special functions of mathematics (the Gamma function, Bessel functions, Hermite functions, etc.)

- Learn to test validity of proposed solutions

- Get practice in making conjectures and asking good questions

- Continue to develop higher-order conceptual thinking in mathematics ("mathematical maturity")

**Delivery Method: Remotely accessible**

Prerequisites: MAT 4288 Calculus: A Classical Approach, or any other college-level calculus (by permission of instructor). Contact Andrew McIntyre by email or Slack, before 4000 registration opens, to ask to be added.

Course Level: 4000-level

Credits: 4

M/Th 3:40PM - 5:30PM (Full-term)

Maximum Enrollment: 20

Course Frequency: Every 2-3 years

Categories: All courses , Mathematics , Remotely Accessible

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