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Differential Equations Online Course for Academic Credit

Differential Equations can be best described as "Higher-Level Integration Theory". The simplest Differential Equations have solutions that are simple Integrals as you learned in Calculus II. But very quickly the Differential Equations become more complicated, and so, too, are the solutions. Physicists think of Differential Equations as the equations that get spit out from their analysis of the various physics situations, and thus need to be solved to understand the physics. Unfortunately, most Differential Equations cannot be solved algebraically, but the main focus of classroom/textbook courses is usually to just try to exhaust all of the Differential Equations that can be solved by hand.

Our Differential Equations online course via Distance Calculus @ Roger Williams University takes a different approach: what do these differential equations mean? What do their solutions mean? What do their graphical or numerical solutions mean? Using a power tool like Mathematica, we are not bound by just those differential equations that have hand-calculated solutions, but rather all differential equations are fair game, and we investigate the concepts of Differential Equations from a laboratory-science point of view.

The first course in a study of Differential Equations is often called Ordinary Differential Equations; other names for this course include:

  • Introductory Differential Equations
  • Ordinary Differential Equations
  • A First Course in Differential Equations
  • Single-Variable Differential Equations

Completion of Math 317 - Differential Equations earns 3 academic credit semester hours with an official academic transcript from Roger Williams University, in Providence, Rhode Island, USA, which is regionally accredited by the New England Commission of Higher Education (NECHE), facilitating transfer of credits nationwide to other colleges and universities.

Differential Equations Online Course Introductory Videos

Differential Equations Online Course Introduction

Differential Equations can be thought of as "the task of integration, with (more and more) complications".

Simple algebraic integration of a function f(x) can be re-interpreted in terms of this integral being the solution of a differential equation y' = f(x), and our task is to solve for y - as integration is the "inverse" operation of differentiation, we see that y is the algebraic integral of f(x).

The equation y' = f(x) is the most basic differential equation possible. Quickly we are lead to investigate more complicated forms of equations involving differentiation, for example: y' = y + f(x), which asks: find a function y = y(x) which has the property that its derivative y' is equal to itself y added to a function f(x). Not an easy question before starting the Differential Equations course, but upon completion of this course, such questions - and exponentially more difficult and complex such equations - are answered with skill and understanding.

More than just an "algebraic game involving integrals", the topic of Differential Equations studies not only the algebraic solutions of such equations (when possible!), but also the qualitative understanding of the properties and solutions of these equations.

Traditional Differential Equations courses often are dedicated to learning these "expanded integration techniques" to study the solutions of these equations purely from an algebraic point of view. While this approach has its merits, the types of differential equations encountered "in the real world" (i.e physics, chemistry, engineering, etc) require solution and analysis techniques beyond what is possible via algebra alone.

The course curriculum for our Differential Equations online course is based upon Differential Equations&Mathematica by Carpenter/Davis/Uhl, and utilizes the high-powered computer algebra and graphing system Mathematica™ by Wolfram Research. Engaging the algebraic investigations of the classes of differential equations studied on briefly, the curriculum moves quickly to investigating the much richer concepts accessible via Mathematica™ and its numerical and graphical differential equation solvers, opening up the introductory study of differential equations beyond the traditional textbook on the subject.

Roger Williams University Course Catalog Listing: Math 317 - Differential Equations

MATH 317: Differential Equations [3 credit hours]

Course Description: Studies methods of solution of ordinary differential equations with applications in science and engineering. Extensive use is made of the method of Laplace transforms.

Prerequisite: Calculus II
Detailed Course Syllabus in PDF

Course Materials for Differential Equations Online Course

The Differential Equations curriculum topics include:

  • Linear First Order Differential Equations
    Topics Include:
    • Unforced Equations (Homogenious)
    • Forced Equations (Non-Homogenious)
    • Steady State Solutions
    • Applications to Personal Finance
    • Step Function and Dirac Delta Function
    • Tangent Vectors
    • Initial Conditions
    • Integration Factors
  • Linear Second Order Differential Equations
    Topics Include:
    • Overdamped and Underdamped Oscillators
    • Linear Forced and Unforced Oscillators
    • Homogeneous Equations
    • Inhomogeneous Equations - Variation of Parameters
    • Characteristic Equations
    • Euler’s Formula
    • Impulse Forcing
    • Convolution Integrals Methods
    • Springs and Electrical Charges
    • Resonance
    • Higher Order Equations
  • Laplace Transforms
    Topics Include:
    • Computing Laplace Transforms
    • Converting Differential Equations via Laplace Transforms
    • Inverse Laplace Transforms
    • Introductory Fourier Analysis
    • Fast Fourier Transforms To Approximate Periodic Functions
    • Laplace Transforms and Fast Fourier Fits
  • Graphical Analysis of Differential Equations
    Topics Include:
    • Euler’s Method
    • Flow Plots and Trajectories
    • Phase Lines
    • Predator-Prey Model
    • Logistic Harvesting
    • Bifurcation Points
    • Sensitivity to Initial Conditions
  • Non-Linear First Order Differential Equations
    Topics Include:
    • Autonomous Equations
    • Non-Autonomous and Other Equation Types
    • Multiple Phase Lines
    • Bifurcation Plots
    • Separation of Variables Solving Method
  • Linear Systems of Differential Equations
    Topics Include:
    • Flows, Trajectories, and Vector Fields
    • Conversion Between Higher Order ODEs and Linear Systems
    • Eigenvalues, Eigenvectors, and Classification of Solutions
  • Differential Equations Online Course Credit: 3 semester credit hours

High School Students - Differential Equations Course

Many advanced high school students who have completed their AP Calculus AB and BC courses, which are equivalent to Calculus I and Calculus II, respectively, often find themselves in their senior year of high school with no further courses in mathematics to take at their high school.

There are no further AP Calculus courses beyond the AP Calculus BC (Calculus II) course.

An excellent next academic move for these advanced high school mathematics students is to take one or more of the following Distance Calculus courses:

  • Math 351 - Multivariable Calculus
  • Math 317 - Differential Equations
  • Math 331 - Linear Algebra
  • Math 315 - Probability Theory (Calculus-Based Statistics)
Unlike the AP Calculus program, these courses do not have "make-or-break standardized exam" to pass, but rather earn real university academic credits through Roger Williams University for these advanced high school students.

Here is a video discussing some options for these advanced high school students.

Advanced High School Math Plan for Senior Year

After AP Calculus

Differential Equations Online Course Credit: 3 semester university-level credit hours

What are Ordinary Differential Equations?

"Ordinary" means that the functions we are studying are functions of one variable. Usually: y = f(x)

Historically, this course was called "Ordinary Differential Equations", often abbreviated as ODE, and you will see ODE used interchangeably with DE to abbreviate the term "Differential Equations".

What is a non-Ordinary Differential Equation? Those are called Partial Differential Equations, and they involve looking at equations involving partial derivatives of functions of two or more variables, like this: z = f(x,y).

Partial Differential Equations is usually a junior-level mathematics course that all mathematics majors, and many physics and engineering majors, will take to learn about the next steps in the study of differential equations.

The term "ordinary" in Ordinary Differential Equations helps you identify the course level of the Differential Equations course you might be required to take. This Differential Equations online course - Math 317 - is the first course on ordinary differential equations.

Colloquially, you may hear other students refer to this course as "Diff-E-Q" or "DiffyQ".

There is also a higher, junior level course on Differential Equations - usually referred to "Advanced Differential Equations", or "Intermediate Differential Equations", which does continue the study of ordinary differential equations, but at a more advanced level, and quite distinct from the partial differential equations courses. Our Math 317 - Differential Equations is not considered to be one of these higher junior-level and above courses.

Differential Equations Examples of the Curriculum

Below are some PDF "print outs" of a few of the Mathematica™ notebooks from Differential Equations&Mathematica by Carpenter/Davis/Uhl. Included as well is an example homework notebook completed by a student in the course, demonstrating how the homework notebooks become the "common blackboards" that the students and instructor both write on in their "conversation" about the notebook.

That Looks Like Programming Code!

Yes, Mathematica™ is a syntax-based computer algebra system - i.e. the instructions to generate the graphs and computations look like a programming language code (which it is).

This course is not a course on programming. We do not teach programming, nor do we expect the students to learning programming, or even to know anything about programming. The mathematics is what is important in this course, not the code.

With that tenet in mind, the authors of the Differential Equations&Mathematica courseware have designed the explanation notebooks (Basics & Tutorials) and the homework notebooks (Give It a Try) in such a way as to make it easy to Copy/Paste from the explanations into the homework notebooks, and make minor changes (obvious ones) to produce the desired similar (but different) output. In this way, we are able to stick strictly to the mathematics at hand, and deal with the programming code as minimally as possible.

sample mathematica notebook

Differential Equations Online Course Credit: 3 semester university-level credit hours

Distance Calculus - Student Reviews

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Date Posted: Mar 16, 2020
Review by: Malia K.
Courses Completed: Applied Calculus
Review: Course was good and fast. I don't like math so I can't say it was fun or anything. Grader was very nice. Software was ok.
Transferred Credits to: University of Maine

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Date Posted: Feb 25, 2020
Review by: Jessica M.
Courses Completed: Applied Calculus
Review: I highly recommend this course. I started the Kennedy School at Harvard with a last-minute admission, but my application required the Liberal Arts calculus course, so I had to finish the course in 3 weeks. Diane was an awesome instructor! The class was surprisingly interesting. If you need to take calculus fast, this is the program to use.
Transferred Credits to: Kennedy School of Government, Harvard University

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Date Posted: Feb 28, 2020
Review by: Teddy M.
Courses Completed: Precalculus, Calculus I
Review: Pros: once you get going, you can go really fast. The visual textbook is pretty cool. The instructors were very responsive. Cons: the movies are great, but the software crashes more than it should. Sometimes it is just a hassle doing things in the software instead of on paper, but once I got used to the software, it was ok.
Transferred Credits to: Texas Christian University

Frequently Asked Questions

Yes. In some ways Differential Equations is the next course after Calculus II, basically the next course in integration theory, now made more complicated by having more involved equations of derivatives of functions.

The term "Ordinary" simply refers to these functions involving a single variable. The study of differential equations with functions of 2 or more variables is called Partial Differential Equations, because they involve partial derivatives.

Yes. Surprisingly, Differential Equations is usually not required as a prerequisite for Data Science programs, but even though not technically on the requirement list, it is a good idea to take all of the Calculus sequence courses in preparation for a Data Science certificate or degree.

No. Calculus II is very much a prerequisite for Differential Equations, and not a nominal prerequisite. Differential Equations relies very strongly on the mastery of the Calculus II content.

Yes, All Distance Calculus courses are offered through Roger Williams University in Providence, Rhode Island, USA, which is regionally accredited (the highest accreditation) through New England Commission of Higher Education (NECHE).