differential equations coursework

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Introduction to Differential Equations

Interactive course.

• Intermediate
• 2 Certifications

Estimated Time : 12 h

Course Level : Intermediate

Requirements : This course requires no prior knowledge of Mathematica or the Wolfram Language. Prerequisites for differential equations include calculus and linear algebra.

Certification Levels : Completion Level 1

A comprehensive introduction to fundamental concepts and solution methods for differential equations, including video lessons and interactive notebooks. Follow along with the examples in the Wolfram Cloud and use the material to prepare for courses in natural science, engineering, economics and other fields. The course starts with a discussion of direction fields and methods for solving first-order differential equations, followed by the study of second-order equations and their applications, and then moves on to solving systems of differential equations. Problem sessions, exercises and quizzes are provided for self-paced assessment. Earn a certificate by watching all lesson and problem session videos and completing the quizzes with a passing grade. Level I certification in differential equations is awarded to those who meet the completion requirements and also pass the course final exam.

Featured Products & Technologies: Wolfram Language (available in Mathematica and Wolfram|One)

You'll Learn To

• Visualize direction fields in the plane
• Compute series solutions of differential equations
• Solve linear systems with constant coefficients
• Find exact solutions of differential equations
• Use Laplace transforms for solving initial value problems
• Apply differential equations to real-world problems

Certifications Available

Completion certificate.

Certify your completion of this course by watching course videos and passing the auto-graded quizzes.

Level 1 Certification

Pass the auto-graded quizzes and final exam to certify your proficiency in differential equations.

About This Interactive Course

It's free and easy to get started with open interactive courses using the Wolfram Cloud—sign in with your Wolfram ID or create one. No plan is required. This interactive course includes video lessons, exercises, problem sessions, quizzes, a final exam and a scratch notebook, all in an easy-to-use interface. From the interactive course, click Track My Progress to chart your certification progress as you go. Recommended best practice for completing this interactive course is to start with Lesson 1 and progress through the video lessons, exercises and problems, taking each quiz in the order it appears in the table of contents.

Ask questions or start a discussion about related topics on Wolfram Community

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Introduction to differential equations

Course description

Course content, course reviews.

Differential equations are any equations that include derivatives and arise in many situations. This free course, Introduction to differential equations, considers three types of first-order differential equations. Section 1 introduces equations that can be solved by direct integration and section 2 the method of separation of variables. Section 3 looks at applications of differential equations for solving real world problems. Section 4 introduces the integrating factor method for solving linear differential equations. The final two sections summarise and revise the methods introduced in the previous sections and describe various other approaches to finding solutions of first-order differential equations and to understanding the behaviour of the solutions.

Course learning outcomes

After studying this course, you should be able to:

• recognise differential equations that can be solved by each of the three methods – direct integration, separation of variables and integrating factor method – and use the appropriate method to solve them
• use an initial condition to find a particular solution of a differential equation, given a general solution
• check a solution of a differential equation in explicit or implicit form, by substituting it into the differential equation
• understand the terms 'exponential growth/decay', 'proportionate growth rate' and 'doubling/halving time' when applied to population models, and the terms 'exponential decay', 'decay constant' and 'half-life' when applied to radioactivity
• solve problems involving exponential growth and decay.

First Published: 27/09/2017

Updated: 12/04/2018

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Ordinary Differential Equations and Linear Algebra - Part 1

A self-paced, comprehensive course to prepare you for the AP Physics 1 exam.

DESCRIPTION

Study of ordinary differential equations (e.g., solutions to separable and linear first-order equations and to higher-order linear equations with constant coefficients, systems of linear differential equations, the properties of solutions to differential equations) and linear algebra (e.g., vector spaces and solutions to algebraic linear equations, dimension, eigenvalues, and eigenvectors of a matrix), as well as the application of linear algebra to first-order systems of differential equations and the qualitative theory of nonlinear systems and phase portraits.

Any questions? Please e-mail [email protected]

PROFESSOR INFORMATION

Elementary Differential Equations Online College Course

UND's differential equations online course covers the solution of elementary differential equations by elementary techniques.

Register Now

Why Take the Elementary Differential Equations Online Course?

This course covers methods of finding solutions to the First Order and second order differential equations. Laplace transforms is introduced to solve initial value problems when a forcing function with jump discontinuity is involved. The theory of linear systems is discussed in normal form and present matrix methods for solving such systems.

This course has 20 lessons and 4 exams, proctored by ProctorU Live+ . You may work at your own pace and submit up to 3 items per week, taking as little as 8 weeks and a maximum of 9 months to complete the course. In addition to the 8 weeks, please allow an additional 5-7 business days to process registration, and 3-5 days for the final grade to appear on your transcript.

The lesson topics include:

• Solutions and Initial Value Problems
• Geometrical or Graphical Consideration (Isoclines)
• Separable Equations
• Linear First Order Differential Equations
• Exact Differential Equations
• Homogeneous Linear Equations
• Auxiliary Equations with Complex Roots
• Method of Undetermined Coefficients
• Superposition and Nonhomogeneous Equations
• Variation of Parameters
• Variable coefficient equations
• Laplace Transforms
• Properties of the Laplace Transforms
• Inverse Laplace Transform and Solving Initial Value Problems
• Transforms of Discontinuous and Periodic Functions and Convolutions
• Matrices and Vectors
• Linear Systems
• Homogeneous Linear System with Constant Coefficients
• Complex Eigenvalues
• Nonhomogeneous Linear Systems

Each lesson (except for the exams) contains lesson objectives and to-do list, Instructional notes<, a brief introductory lecture, and an assessment.

The course syllabus provides the course description and objectives, lesson topics, assignment information, grading criteria, and more.

Elementary Differential Equations Course Requirements

The textbook and student solutions manual are both required for this course.

Nagle, R. K., Saff, E. B., & Snider, A. D. (2018). Fundamentals of Differential Equations (9th Edition) Pearson Publishing.

• ISBN:10-0321977068
• ISBN 13: 978-0-321-97706-9
• We will be covering chapters 1, 2, 4, 7, and 9.

Student's Solutions Manual for Fundamentals of Differential Equations 8e and Fundamentals of Differential Equations and Boundary Value Problems 6e 6th edition

• ISBN-13: 9780321748348
• ISBN-10: 03217488344
• Victor Maymeskul

How will the course appear on my transcript?

You may enroll at any time and have up to 9 months to complete this online course. The credits earned will be recorded on your UND transcript based on the date you registered for the course. It will appear on your transcript in the same way as a course taken during a regular semester. There is no indication that the course was taken online or that you completed it at your own pace. 

Why Take Online Classes at UND?

Here are a few reasons why you should take an online enroll anytime course at UND:

• Great customer service – Our registration team is ready to answer questions quickly so you can focus on your coursework.
• Affordable – UND's enroll anytime courses are priced  at North Dakota's affordable, in-state tuition rate.
• Accredited – UND is accredited by the Higher Learning Commission .
• Easily transfer credits – Transferring credits is always at the discretion of the institution to which the credits are being transferred. In general, credits from schools/universities that are regionally accredited by the Higher Learning Commission transfer to other regionally accredited institutions. UND's online courses appear on your UND transcript in the same way as other courses.

Flexible 100% Online Course

You'll take this online course at your own pace. Some students thrive in this environment, while other students may struggle with setting their own deadlines. If you have successfully taken an independent study or correspondence course previously, UND’s enroll anytime courses may be right for you. Still not sure? Take our online quiz to help determine if online enroll anytime courses are right for you.

Course information including tuition, technology requirements, textbooks, lessons and exams is subject to change without notice.

Complete your college degree with UND’s top ranked communication, social science or general studies programs. You can take classes 100% online with enroll anytime courses.

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A6: Differential Equations 2 (2024-25)

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A framework for solving parabolic partial differential equations

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Computer graphics and geometry processing research provide the tools needed to simulate physical phenomena like fire and flames, aiding the creation of visual effects in video games and movies as well as the fabrication of complex geometric shapes using tools like 3D printing.

Under the hood, mathematical problems called partial differential equations (PDEs) model these natural processes. Among the many PDEs used in physics and computer graphics, a class called second-order parabolic PDEs explain how phenomena can become smooth over time. The most famous example in this class is the heat equation, which predicts how heat diffuses along a surface or in a volume over time.

Researchers in geometry processing have designed numerous algorithms to solve these problems on curved surfaces, but their methods often apply only to linear problems or to a single PDE. A more general approach by researchers from MIT’s Computer Science and Artificial Intelligence Laboratory (CSAIL) tackles a general class of these potentially nonlinear problems.  In a paper recently published in the Transactions on Graphics journal and presented at the SIGGRAPH conference, they describe an algorithm that solves different nonlinear parabolic PDEs on triangle meshes by splitting them into three simpler equations that can be solved with techniques graphics researchers already have in their software toolkit. This framework can help better analyze shapes and model complex dynamical processes.

“We provide a recipe: If you want to numerically solve a second-order parabolic PDE, you can follow a set of three steps,” says lead author Leticia Mattos Da Silva SM ’23, an MIT PhD student in electrical engineering and computer science (EECS) and CSAIL affiliate. “For each of the steps in this approach, you’re solving a simpler problem using simpler tools from geometry processing, but at the end, you get a solution to the more challenging second-order parabolic PDE.” To accomplish this, Da Silva and her coauthors used Strang splitting, a technique that allows geometry processing researchers to break the PDE down into problems they know how to solve efficiently.

First, their algorithm advances a solution forward in time by solving the heat equation (also called the “diffusion equation”), which models how heat from a source spreads over a shape. Picture using a blow torch to warm up a metal plate — this equation describes how heat from that spot would diffuse over it. 
This step can be completed easily with linear algebra.

Now, imagine that the parabolic PDE has additional nonlinear behaviors that are not described by the spread of heat. This is where the second step of the algorithm comes in: it accounts for the nonlinear piece by solving a Hamilton-Jacobi (HJ) equation, a first-order nonlinear PDE.  While generic HJ equations can be hard to solve, Mattos Da Silva and coauthors prove that their splitting method applied to many important PDEs yields an HJ equation that can be solved via convex optimization algorithms. Convex optimization is a standard tool for which researchers in geometry processing already have efficient and reliable software. In the final step, the algorithm advances a solution forward in time using the heat equation again to advance the more complex second-order parabolic PDE forward in time.


Among other applications, the framework could help simulate fire and flames more efficiently. “There’s a huge pipeline that creates a video with flames being simulated, but at the heart of it is a PDE solver,” says Mattos Da Silva. For these pipelines, an essential step is solving the G-equation, a nonlinear parabolic PDE that models the front propagation of the flame and can be solved using the researchers’ framework.

The team’s algorithm can also solve the diffusion equation in the logarithmic domain, where it becomes nonlinear. Senior author Justin Solomon, associate professor of EECS and leader of the CSAIL Geometric Data Processing Group, previously developed a state-of-the-art technique for optimal transport that requires taking the logarithm of the result of heat diffusion. Mattos Da Silva’s framework provided more reliable computations by doing diffusion directly in the logarithmic domain. This enabled a more stable way to, for example, find a geometric notion of average among distributions on surface meshes like a model of a koala. Even though their framework focuses on general, nonlinear problems, it can also be used to solve linear PDE. For instance, the method solves the Fokker-Planck equation, where heat diffuses in a linear way, but there are additional terms that drift in the same direction heat is spreading. In a straightforward application, the approach modeled how swirls would evolve over the surface of a triangulated sphere. The result resembles purple-and-brown latte art.

The researchers note that this project is a starting point for tackling the nonlinearity in other PDEs that appear in graphics and geometry processing head-on. For example, they focused on static surfaces but would like to apply their work to moving ones, too. Moreover, their framework solves problems involving a single parabolic PDE, but the team would also like to tackle problems involving coupled parabolic PDE. These types of problems arise in biology and chemistry, where the equation describing the evolution of each agent in a mixture, for example, is linked to the others’ equations.

Mattos Da Silva and Solomon wrote the paper with Oded Stein, assistant professor at the University of Southern California’s Viterbi School of Engineering. Their work was supported, in part, by an MIT Schwarzman College of Computing Fellowship funded by Google, a MathWorks Fellowship, the Swiss National Science Foundation, the U.S. Army Research Office, the U.S. Air Force Office of Scientific Research, the U.S. National Science Foundation, MIT-IBM Watson AI Lab, the Toyota-CSAIL Joint Research Center, Adobe Systems, and Google Research.

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Related links.

• Leticia Mattos Da Silva
• Justin Solomon
• Computer Science and Artificial Intelligence Laboratory (CSAIL)
• MIT-IBM Watson AI Lab
• Department of Electrical Engineering and Computer Science

Related Topics

• Computer science and technology
• Artificial intelligence
• Computer graphics
• Electrical Engineering & Computer Science (eecs)

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Browse Course Material

Course info, instructors.

• Prof. Haynes Miller
• Prof. Arthur Mattuck

Departments

• Mathematics

As Taught In

• Differential Equations
• Linear Algebra

Learning Resource Types

Course meeting times.

Lectures: 3 sessions / week, 1 hour / session

Recitations: 2 sessions / week, 1 hour / session

Prerequisites/Corequisites

18.03 Differential Equations has 18.01 Single Variable Calculus as a prerequisite. 18.02 Multivariable Calculus is a corequisite, meaning students can take 18.02 and 18.03 simultaneously.

Edwards, C., and D. Penney. Elementary Differential Equations with Boundary Value Problems . 6th ed. Upper Saddle River, NJ: Prentice Hall, 2003. ISBN: 9780136006138.

Note: The 5th Edition (ISBN: 9780131457744) will serve as well.

Students also need two sets of notes “ 18.03: Notes and Exercises ” by Arthur Mattuck, and “ 18.03 Supplementary Notes ” by Haynes Miller.

Description

This course is a study of Ordinary Differential Equations (ODE’s), including modeling physical systems.

Topics include:

• Solution of First-order ODE’s by Analytical, Graphical and Numerical Methods;
• Linear ODE’s, Especially Second Order with Constant Coefficients;
• Undetermined Coefficients and Variation of Parameters;
• Sinusoidal and Exponential Signals: Oscillations, Damping, Resonance;
• Complex Numbers and Exponentials;
• Fourier Series, Periodic Solutions;
• Delta Functions, Convolution, and Laplace Transform Methods;
• Matrix and First-order Linear Systems: Eigenvalues and Eigenvectors; and
• Non-linear Autonomous Systems: Critical Point Analysis and Phase Plane Diagrams.

The lecture period is used to help students gain expertise in understanding, constructing, solving, and interpreting differential equations. Students must come to lecture prepared to participate actively. At the first recitation, students are given a set of flashcards to bring to each lecture. They are used during class sessions to vote on answers to questions posed occasionally in the lecture. In case of divided opinions, a discussion follows. As a further element of active participation in class, students will often be asked to spend a minute responding to a short feedback question at the end of the lecture.

Recitations

These small groups meet twice a week to discuss and gain experience with the course material. Even more than the lectures, the recitations involve active participation. The recitation leader may begin by asking for questions or hand out problems to work on in small groups. Students are encouraged to ask questions early and often. Recitation leaders also hold office hours.

Another resource of great value to students is the tutoring room. This is staffed by experienced undergraduates. Extra staff is added before hour exams. This is a good place to go to work on homework.

The Ten Essential Skills

Students should strive for personal mastery over the following skills. These are the skills that are used in other courses at MIT. This list of skills is widely disseminated among the faculty teaching courses listing 18.03 as a prerequisite. At the moment, 140 courses at MIT list 18.03 as a prerequisite or a corequisite.

• Model a simple system to obtain a first order ODE. Visualize solutions using direction fields and isoclines, and approximate them using Euler’s method.
• Solve a first order linear ODE by the method of integrating factors or variation of parameter.
• Calculate with complex numbers and exponentials.
• Solve a constant coefficient second order linear initial value problem with driving term exponential times polynomial. If the input signal is sinusoidal, compute amplitude gain and phase shift.
• Compute Fourier coefficients, and find periodic solutions of linear ODEs by means of Fourier series.
• Utilize Delta functions to model abrupt phenomena, compute the unit impulse response, and express the system response to a general signal by means of the convolution integral.
• Find the weight function or unit impulse response and solve constant coefficient linear initial value problems using the Laplace transform together with tables of standard values. Relate the pole diagram of the transfer function to damping characteristics and the frequency response curve.
• Calculate eigenvalues, eigenvectors, and matrix exponentials, and use them to solve first order linear systems. Relate first order systems with higher-order ODEs.
• Recreate the phase portrait of a two-dimensional linear autonomous system from trace and determinant.
• Determine the qualitative behavior of an autonomous nonlinear two-dimensional system by means of an analysis of behavior near critical points.

The Ten Essential Skills is also available as a ( PDF ).

Each homework assignment has two parts: a first part drawn from the book or notes, and a second part consisting of problems which will be handed out. Both parts are keyed closely to the lectures. Students should form the habit of doing the relevant problems between successive lectures and not try to do the whole set the night before they are due.

There are 3 one-hour exams held during lecture session and a three-hour comprehensive final examination.

The final grade will be based on a cumulative total of 885 points computed as follows:

ODE Manipulatives (“Mathlets”)

This course employs a series of specially written Java™ applets, or Mathlets . They are used in lecture occasionally, and each problem set contains a problem based around one or another of them.

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