What to cover in a differential equations module?

[matqkks]

I will have to teach a first course in differential equations. What should I cover in this module? For example, in most books they have Laplace Transforms which is fine but I would not use LT to solve differential equations.

I want to write a course so that it motivates students and has an impact. What topics and what is the most motivating way to introduce differential equations? I want a well structured and a practical approach to differential equations. It is for mathematics and physics students.

 

[R.M.]

Is this a semester course? Full year course? Short (9 weeks or less)? For my answer, I assumed a semester course.

My 2 cents/suggestions:
- Do a google search for some sample syllabi to see what topics other instructors cover. This will give you an idea of what topics are "important", what topics are "nice-if-we-have-time", and what topics are "Hey-I-never-thought-of-that-topic-before". Will also give you different ideas for various topic sequence combinations.
- I've only taught a short chapter on differential equations as part of a calculus course, but if I were ever to teach a more extensive course I would start with Paul's Online Math Notes: https://tutorial.math.lamar.edu/Classes/DE/DE.aspx I've used his website for a lot of supplemental material with my calculus students.
- Think back to your own experience as a student in that course. What topic(s) did you feel were important? What topic(s) did you feel needed more or less treatment?

 

[farhan]

A module on differential equations typically covers the following topics:

1. First-order ordinary differential equations (ODEs):

• Separable equations
• Linear equations
• Exact equations
• Bernoulli equations
• Integrating factors

1. Second-order ODEs:

• Homogeneous equations with constant coefficients
• Nonhomogeneous equations with constant coefficients
• Method of undetermined coefficients
• Variation of parameters
• Cauchy-Euler equations

1. Systems of ODEs:

• Linear systems
• Nonlinear systems
• Phase plane analysis

1. Laplace transforms:

• Definition and basic properties
• Inverse Laplace transforms
• Laplace transforms of derivatives and integrals
• Laplace transforms of ODEs
• Convolution theorem

1. Partial differential equations (PDEs):

• Classification of PDEs
• Method of characteristics
• Separation of variables
• Fourier series and Fourier transforms
• Heat equation
• Wave equation
• Laplace's equation

1. Numerical methods for solving ODEs and PDEs:

• Euler's method
• Runge-Kutta methods
• Finite difference methods
• Finite element methods

In addition to these topics, a differential equations module may also cover applications of differential equations in various fields such as physics, engineering, economics, and biology.