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.

Course

DMAT 321 - Computational Differential Equations

Credits

3 Semester Credit Hours

Delivery

Fully Online, Asynchronous, Self-Paced

Syllabus

Download PDF Syllabus

Click Here to Enroll in DMAT 321

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 DMAT 321 - Computational 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 adapted from Differential Equations & Mathematica by Carpenter/Davis/Uhl - part of the Calculus & Mathematica courseware series - and uses LiveMath as the computer algebra and graphing system. Engaging the algebraic investigations only briefly, the curriculum moves quickly to investigating the much richer concepts accessible via LiveMath's numerical and graphical differential equation solvers, opening up the introductory study of differential equations far beyond the traditional textbook on the subject.

A Laboratory Approach to Differential Equations

Most classroom Differential Equations courses follow a "cookbook recipe" approach: here is how you solve this kind of equation, here is how you solve that kind. Memorize the procedures, run them on exam day. That has merits, but it leaves out something essential about the subject.

DMAT 321 is built differently. Using LiveMath as our laboratory tool, the course feels like working in a real laboratory - the way a biology student uses a microscope, or a chemistry student uses beakers and reagents. LiveMath is the instrument; differential equations are the specimens. You solve them algebraically when possible, then look at the solutions numerically and graphically to understand the nature of the equation, not just its formula.

Three Perspectives: Algebraic, Numerical, Graphical

Most differential equations textbooks dwell almost entirely on the algebraic side. Our course explores three perspectives in parallel:

Algebraic - When a closed-form algebraic solution exists, we find it and work with it.
Numerical - When algebra fails, numerical solvers approximate the solutions to any precision.
Graphical - Phase plots, slope fields, flow plots, and 3D visualizations show the behavior of solutions even when no formula is available.

The result: we are not restricted to those differential equations that happen to have hand-solvable forms. The differential equations that engineers, physicists, and scientists encounter in the real world rarely fit those neat algebraic boxes. With numerical and graphical tools at hand, we can study them anyway.

What You'll Cover in DMAT 321

The course progresses through the standard catalog of differential-equation types and techniques:

First-order linear differential equations
Second-order linear differential equations
Higher-order linear differential equations
Nonlinear differential equations
Systems of differential equations in 2D - phase plane and flow plots
Systems of differential equations in 3D - the dramatic flow visualizations you see on this page (Lorenz attractor, Duffing oscillator, non-autonomous flows)

Preparation for Engineering and Physics Applications

Will DMAT 321 prepare you for engineering and physics work that uses differential equations? Absolutely. But the preparation goes well beyond memorizing algebraic techniques.

You leave DMAT 321 with full command of the subject. When a differential equation appears in your other coursework or research, you do not stare at it wondering if there is a formula somewhere. You know how to study it: how to analyze it, how to graph its solutions, how to set up numerical investigation, and which algebraic techniques (if any) apply. That is a much stronger working toolkit than a textbook formula sheet.

DMAT 321 vs DMAT 322 Honors

DMAT 321 is a 3-credit course - a notch faster and lighter than the other sophomore-level courses in the catalog. If you are looking for a more expansive honors-level treatment of the same material, take a look at DMAT 322 - Honors Differential Equations: same laboratory approach, same three-perspective philosophy, with an additional credit hour and using Mathematica as the computer algebra system.

RWU Course Catalog - DMAT 321 • Computational Differential Equations

Course

DMAT 321

Course Title

Computational Differential Equations

Transcript Title

Differential Equations

Credits

3 Semester Credit Hours

Description

A first course in the study of differential equations with emphasis on modern software computational techniques with geometrical and qualitative interpretations. Topics include first, second, and higher-order ordinary differential equations, analysis of forcing functions, Laplace Transforms, convolution integral techniques, Fast Fourier Transforms and data approximations, systems of differential equations, classical algebraic solution methods, power series solutions.

Prerequisite

Successful completion (C- or higher) of Calculus II or equivalent, or consent of instructor.

E-Textbook

Differential Equations & Mathematica by Davis/Porta/Uhl

Software

Mathematica

Syllabus PDF

Download Detailed Syllabus (PDF)

DMAT 321 - Learning Outcomes

To understand the core construction of the differential equation, and its classification parts
To understand the role of the forcing function in differential equations
To understand, observe, and compute the steady state solutions for a differential equation
To understand, observe, and compute solutions of differential equations with a variety of forcing functions, including DiracDelta, step, oscillatory, and others
To understand, observe, and compute solutions of differential equations with a variety of forcing functions, including DiracDelta impulses, step, oscillatory, et al.
To understand and compute solutions of second order differential equations oscillators and how forcing functions affect their solutions
To understand and compute manual solutions of first and second order differential equations using classical techiques
To understand and compute with the Laplace Transform method
To understand and compute graphical and numerical solution methods of differential equations
To understand and compute solutions of linear systems of differential equations
To understand and compute polynomial approximations to solutions of differential equations

DMAT 321 - Syllabus of Topics

1. Getting Started

1.1. Email and Chat

1.2. Learning About the Course

1.3. Required Hardware

1.4. Software Fundamentals

2. Exponential Differential Equations

2.1. Unforced DEs

2.2. Forced DEs

2.3. Steady State

2.4. Personal Finance

2.5. Step Function and Dirac Delta Function

2.6. Tangent Vectors

2.7. Initial Conditions

2.8. Integration Factors

3. Second-Order Differential Equations

3.1. Overdamped and Underdamped Oscillators

3.2. Linear Forced and Unforced Oscillators

3.3. Homogeneous and Inhomogeneous Equations

3.4. Convolution Method

3.5. Characteristic Equations

3.6. Euler�s Formula

3.7 Impulse Forcing

3.8 Dirac Delta Convolutions

3.9 Springs and Electrical Charges

3.10 Higher Order Equations

4. Laplace Transforms

4.1 Laplace Transforms of First and Second Order Equations

4.2 Fourier Analysis and Fourier Fit Approximations

5. Graphical Analysis of Differential Equations

5.1 Euler�s Method

5.2 Flow Plots and Trajectories

5.3 Preditor-Prey Model

5.4 Logistic Harvesting

6. First-Order Differential Equations

6.1 Autonomous Equations

6.2 Non-Autonomous Equations

6.3 Separation of Variables Solving Method

7. Systems of Differential Equations

7.1 Flows and Trajectories

7.2 Conversion Between Higher Order ODEs and Systems

7.3 Relationship to Eigenvalues and Eigenvectors

8. Power Series Solutions of Differential Equations

8.1 Recursion Relations

8.2 Comparing Series Solution to Numerical Solution

8.3 Barriers

Legacy Course - DMAT 317 • Differential Equations

In 2023, Distance Calculus introduced a new catalog of courses. New DMAT 321 = Old DMAT 317 = Old MATH 317

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. (3 credits)

Syllabus PDF

Legacy Detailed Course Syllabus (PDF)

Common Completion Timelines for DMAT 321 - Differential Equations

Hours/Week

Math Skills

Dedication

Completion

Advisory

5-10 hrs/wk

Weaker

1-2 hrs/day

14 weeks

Reasonable

7-12 hrs/wk

Modest

2-3 hrs/day

10 weeks

Reasonable

10-15 hrs/wk

Stronger

3-4 hrs/day

7 weeks

Reasonable

15-20 hrs/wk

Strong

5-6 hrs/day

5 weeks

Stretched

20-25 hrs/wk

Strong

5-7 hrs/day

4 weeks

Stretched

25-35 hrs/wk

Strong

6-8 hrs/day

3 weeks

World's Record

Time commitments are important for success in an online Differential Equations course for college credit from Distance Calculus. There are no fixed due dates in the Distance Calculus online courses, so it is important that students instead set their schedules for a dedicated amount of time towards the coursework.

It is also very important to consider that going faster through a course is DIRECTLY DEPENDENT upon your math skill level, and your successful engagement of the course. We require that you complete the course in a Mastery Learning format. If you are struggling with the course content, or trying to go too fast where the quality of your submitted work is suffering, then the instructors will force a slow-down of your progress through the course, even if you have fixed deadlines.

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:

DMAT 355 - Multivariable Calculus
DMAT 321 - Differential Equations
DMAT 335 - Linear Algebra
DMAT 311 - 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 - DMAT 321 - 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 DMAT 321 - 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.

Sample Curriculum PDFs

Basics - Steady State Solutions

View PDF

Homework - Convolution Integral

View PDF

Instructor Help Movie

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 learn 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.

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

Distance Calculus Referenced Colleges & Universities (29 Years - 393+ Institutions)

Distance Calculus students have transferred course credits to these colleges and universities:

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Distance Calculus - Student Reviews

Allie★★★★★

Posted: May 17, 2025

Courses Completed: Calculus I

This Calculus class saved my life! I needed to take this class to be able to secure a spot in my grad school program, and this was perfect for me - super flexible to my schedule, the teaching was great, and the material was super clear.

I felt like I really learned Calculus and also really enjoyed it!

William Williams★★★★★

Posted: Jan 19, 2020

Courses Completed: Linear Algebra, Probability Theory

I have difficulty learning calculus based math, akin to dyslexia when examining the symbolic forms, equations, definitions, and problems. Mathematica based calculus courses allowed me to continue with my studies because of the option of seeing the math expressed as a programming language for which I have no difficulty in interpreting visually and the immediate feedback of graphical representations of functions, equations, or data makes a huge impact on understanding. Mathematica based calculus courses should be the default method of teaching Calculus everywhere.

Email: wf.williamster@gmail.com

Transferred Credits To: Thomas Edison State College

Colin Donahoe★★★★☆

Posted: May 17, 2025

Courses Completed: Calculus II

Dr. Curtis's lessons were well taught, and livemath wasn't too bad once you understood it. Some of the assignments were pretty difficult and required some more explanation, which was hard to get outside of the given lessons.

Email: colindonahoe4@gmail.com

Transferred Credits To: Saint Joseph's Univesity

Lucas L.★★★★★

Posted: Jun 25, 2026

Courses Completed: Multivariable Calculus

The professor as well as the TAs give great feedback when you need help with problems and the videos are great at explaining concepts. Return time on work is good and the work is not too much to handle.

Transferred Credits To: University of Wisconsin

Hari K.★★★★☆

Posted: Jun 24, 2026

Courses Completed: Linear Algebra

This course gives a perspective on Linear algebra that no traditional course does. I’d say i gained much more intuition for this subject from the DC course than my friends who took traditional courses elsewhere. As a cs major, this version of learning with visualization has helped me a lot in understand ML models. However the course doesn’t have videos for the last 2 chapers so i had to self learn with the mathematica notebooks. Response times are a little slow but since it’s a remote class, i guess it’s justified. Overall amazing course and definitely take this over traditional lin alg classes.

Julia★★★★★

Posted: Jun 24, 2026

Courses Completed: Calculus I

As a full-time business owner completing an Executive MBA, I needed to satisfy a calculus prerequisite without putting my work on hold. Distance Calculus made that possible. The fully self-paced structure let me work early mornings and weekends around an unpredictable schedule, which a fixed-semester classroom course never would have allowed.
The course covered the core business calculus material thoroughly — derivatives, optimization, integration techniques including u-substitution, the Fundamental Theorem of Calculus, improper integrals, and numerical methods. The LiveMath computer algebra environment was central to the experience: it forced me to build each step explicitly rather than just arriving at an answer, which actually deepened my understanding of the mechanics.
Communication through the student portal was responsive when I had questions. For working professionals who need a rigorous, accredited calculus course on a flexible timeline, I'd recommend it.

Transferred Credits To: MIT Ebma

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Frequent Questions

Frequently Asked Questions

Does Differential Equations Have Calculus II As A Prerequisite?

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.

Why Are These Called Ordinary Differential Equations

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.

Is Differential Equations Good for Data Science?

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.

Can I Take Differential Equations Without Calculus II?

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.

Is Differential Equations from Distance Calculus Accredited?

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).