Differential Equations Course Research Articles

PYTHON IN ORDINARY DIFFERENTIAL EQUATIONS LEARNING

Using software in mathematics learning can improve students' soft and hard mathematics skills at the high school and college levels. Therefore, using software in the learning process is important, including in learning Differential Equations. This research examines the use of the Python programming language with Jupyter Lab software and the SymPy library in solving ordinary differential equation problems symbolically in the Differential Equations course. The use of the Python programming language in Differential Equations learning includes solving linear ordinary differential equations of first-order, second-order, higher-order, and the Laplace transform. This research also examines the effect of using Python on the learning outcomes of differential equations of Mathematics Education Study Program students, Study Program Outside the Main Campus, Pattimura University. The population in this quantitative research is all students who programmed differential equations courses in the even semester of the 2023-2024 academic year as many as 19 students. The Python programming language can be used to solve differential equation problems symbolically easily, quickly, and accurately. In addition, using Jupyter Lab makes the process of solving differential equation problems easier and more interactive. Furthermore, t-test results show that the use of Python in learning differential equations can improve students' learning activities and learning outcomes. Using the Python programming language with Jupyter Lab software and the SymPy library can be developed to create teaching materials, textbooks, and reference books for Differential Equations courses.

Perspectives of Experts in the Selection of Contents for a Course of Differential Equations in Engineering

This study investigates the teaching of Differential Equations in engineering education, focusing on expert opinions from diverse institutional backgrounds. The primary objectives were to understand expert perspectives on essential course content, justify their content choices, measure institutional gaps, and highlight the range of professional profiles. Through qualitative interviews with nine experts from academia and industry, we explored their views on the integration of Differential Equations in engineering curricula. The findings reveal a consensus on the importance of basic concepts to support physical models. Notably, experts highlighted variational calculus and energy representations as critical yet underrepresented areas in Differential Equations courses, essential for modeling complex, nonlinear problems. The study also states on the growing importance of computational methods and software in engineering education, advocating for an integrated approach aligning theoretical concepts with real-world applications. The expert opinions revealed a significant institutional influence on content perception, demonstrating a gap between traditional theoretical focus and modern, application-oriented approaches. The study suggests a need for curricular innovation in differential equations education, emphasizing practical applications and computational methods to bridge this gap between the two institutions considered in which the experts are adhered to.

Enhancing students’ learning experience in mathematics class through ChatGPT

This research paper explores the integration of ChatGPT as a tool for interactive learning of numerical methods in a differential equations (DEs) course. DE course is crucial for engineering students to model real-world phenomena. However, many DE courses focus only on analytical solutions and neglect important numerical solutions. To overcome this gap, an active learning approach was adopted, where students were asked to select a real-life phenomenon in their field of interest, find an appropriate model to describe it using DEs, and subsequently solve DE numerically using various numerical methods. The authors allowed students to use ChatGPT to generate codes and learn how to adapt them to their personalized learning needs. The findings derived from the students’ survey responses demonstrated that ChatGPT significantly enhanced their interests and improved their coding skills. Students reported increased engagement and excitement in their learning journey with the integration of ChatGPT. By sharing this experience, we aim to inspire other educators to explore the potential of innovative technologies and instructional methods to enhance their own courses.

Mathematical modelling of hanging rope problem subject to rotation

The hanging rope problem is considered. The rope is subject to rotation with the rotation axis being parallel to the rope. Using a continuum model and the basic principles of dynamics, the differential equation governing the motion is derived. The dynamic equilibrium case without vibrational motion is assumed in deriving the equation. The equation is cast into a dimensionless form first. It is shown that there are two dimensionless parameters affecting the motion, the dimensionless angular velocity parameter and the dimensionless radius of rotation. The differential equation is solved exactly as well as approximately and the effects of parameters on the motion are discussed in detail. The problem can be discussed in a mechanics or differential equation course at the undergraduate level within the context of applications of differential equations to real-world problems.

Advanced Mathematics Course Ideological and Political Integration Strategies and Practice Research

Taking the differential equations course as an example, this paper discusses the integration strategies and practice of ideological and political education in advanced mathematics courses, and analyzes the logicality of the knowledge system of the differential equations course, the advantages of cultivating students' mathematical thinking, and the connotation and extension of ideological and political education. It reveals the lack of ideological and political education in the teaching of the differential equations course, the insufficient exploration of ideological and political education resources, and the monotony of the ideological and political education model. In response to these problems, it proposes to strengthen the integration and innovation of the content of ideological and political education in the differential equations course, improve the ability and quality of teachers' ideological and political education in the differential equations course, and build a diversified ideological and political education model in the differential equations course. Summarizes the importance, optimization direction and strategy of ideological and political education in advanced mathematics courses, and the enlightenment to my country's ideological and political education in advanced mathematics courses.

Teaching differential equations through a mathematical modelling approach: the impact on problem-solving and the mathematical performance of engineering undergraduates

Mathematical modelling is an interlinking process between mathematics and real-world problems that can be applied as a means to increase motivation, develop cognitive competencies, and enhance the ability to transfer mathematical knowledge to other areas of science, such as engineering disciplines. This study was designed to investigate the effect of applying the modelling approach on problem-solving and mathematical performance of engineering undergraduates during a differential equations (DEs) course and compare the results with those of conventional teaching. A quasi-experimental method comprising a pretest-posttest design with a control group was applied in this study. Since the antecedent knowledge of calculus directly influences the ability of students to learn DEs, a placement test was first applied to both groups. A combination of results derived from the placement test and the modelling pretest was used as the covariate in the Analysis of Covariance (ANCOVA), and the modelling and mathematical performances of both groups were eventually examined. The results of this study indicate that apart from the fact that teaching DEs based on the modelling approach can increase the ability to apply mathematical knowledge in the field of engineering, it also can significantly enhance their problem-solving and mathematical performances.

Description of Parameter Variation Learning with Artificial Intelligence and GeoGebra in Students of a Differential Equations Course

In this era of social and technological change, the advent of AI has had great impact on the lives of many people and especially in the educational field, as mentioned in [1], wich prove that the perceptions that students and instructors have about artificial intelligence are very positive to implement them in learning. In this [2] assert that it is possible to have a collaborations between the fields of mathematics and artificial intelligence AI.

Infusing Quantitative Reasoning Skills into a Differential Equation Class in an Urban Public Community College

This research centers on implementing Quantitative Reasoning (QR) within a differential equations course at an urban public community college. As a participant in the Numeracy Infusion for College Educators (NICE) faculty development program, I sought to integrate QR skills into my curriculum. Students in the course were introduced to QR goals using real-world data sets, particularly those related to population growth, which aim to enhance their understanding, sharpen their problem-solving abilities, and cultivate a positive perspective on the real-world relevance of mathematics. Preliminary findings indicate varied levels of QR skill development among students. These results underscore the potential benefits of infusing QR into mathematics courses and provide insights for educators looking to adopt similar strategies in their teaching.

Pull-Back Cars: Vehicles for the Instruction of Differential Equations

This article develops a mathematical model for the motion of toy pull-back cars that can be introduced and revisited throughout an undergraduate differential equations course. This survey of the pull-back car modeling scenario enables students to connect course content with real-world experiences and experimental data, while leaving room for further exploration and advancement of the model.

The Impact and Role of the Ordinary Differential Equations Course on Students’ Ideological and Political Education

The Impact and Role of the Ordinary Differential Equations Course on Students’ Ideological and Political Education

Exploring Students’ Achievements in Differential Equations Courses and Their Feelings Towards Mathematics Courses

Differential equations are a fundamental area of mathematics that is widely used in various fields, including physics, engineering, economics, and more. However, many students find this course to be particularly challenging and struggle to achieve success in differential equations courses. Therefore, this study aims to investigate the relationship between students' achievements in midterm and final exams differential equations courses and their feelings towards mathematics courses, to provide a more comprehensive understanding of the factors that contribute to success in this challenging course. The results of this study indicate a positive correlation between students' achievements in differential equations courses and their feelings toward mathematics. Male students performed slightly better than female students for marks above 50. Additionally, students who enjoyed all the mathematics courses had higher grades in differential equations courses, which was statistically significant. These findings also highlight the importance of midterm exams as preparation before seating for the final exam of differential equations.

Pólya’s Methodology for Strengthening Problem-Solving Skills in Differential Equations: A Case Study in Colombia

The formation of students is integral to education. Strengthening critical thinking and reasoning are essential for the professionals that today’s world needs. For this reason, the authors of this article applied Pólya’s methodology, an initiative based on observing students’ difficulties when facing mathematical problems. The present study is part of the qualitative and quantitative research paradigm and the action research methodology. In this study, the inquiry process was inductive, the sample is non-probabilistic, and the data interpretation strategy is descriptive. As a case study, six students were enrolled onto a differential equations course at the Universidad Autónoma de Bucaramanga. A didactic process was designed using information and communication technologies (ICTs) in five sequences that address first-order differential equation applications. As a result of the pedagogical intervention, problem-solving skills were strengthened. All this was based on asking the right questions, repeated reading, identifying and defining variables, mathematization, communication, and decomposing the problem into subproblems. This research study seeks to set a precedent in the Latin American region that will be the basis for future studies.

Model, analyse and prevent fatal aircraft manoeuvres

On 24 June 1994 at Fairchild Air Force Base, during practice for an air show, a low-flying B-52H aircraft banked its wings vertically and crashed. Emphasizing the activity of modeling drag and gravity, these notes examine the possibility of recovery with several models. First, with algebra, historical data lead to a model where in a free fall near Earth's surface, the distance fallen is proportional to time squared. Calculus then gives an ordinary differential equation to model free fall near Earth's surface. Second, with calculus, historical data lead to various models of drag on objects moving through air. Third, combining models of drag and gravity with Newton's Laws of Motion leads to ordinary differential equations that model free fall in air. Fourth, predictions from such models with ordinary differential equations are consistent with the aircraft crash. Further models examine the possibility of increasing engine thrust to regain the vertical component of lift. All models fit in a first course in differential equations, without requiring any computational machinery. However, numerical experiments show how uninformed use of professional software can produce rounding errors to cause the modelled aircraft to plunge to the bottom of the deepest oceans and shoot up into space.

Love stories in a differential equations classroom

We believe that developing cultural competencies can help students learn mathematics and conversely that learning mathematical content can help students learn about themselves and others. Using frameworks introduced by Rendón (2009) and Gutiérrez (2018), we present a five-part bundle of activities for undergraduate differential equations course instructors, including one pre-activity reflection assignment, three modeling activities, and a final project. The Pre-Activity Assignment engages students to draw on their own personal and/or cultural experiences with the concept of love. Three activities focus on developing revision skills in mathematical modeling and practicing methods of solving systems of ordinary differential equations (ODEs). In these activities, students would collaborate to construct a relationship model consisting of a system of first-order linear ODEs and solve different model iterations with the characteristic polynomial, matrix form, or Laplace transform method. The final project connects the reflections in Pre-Activity Assignment with the skills developed in the three activities by inviting students to create a relationship scenario, model, revise, solve it, and present the conclusions. By engaging in this set of assignments, students connect personal and cultural experiences with the concept of love and perceive themselves and their peers in the curriculum, fostering a sense of belonging and relevance.

Falling Water: A Modeling Story

A standard element of the undergraduate ordinary differential equations course is the topic of separable equations. For instructors of those courses, we present here a series of novel modeling scenarios that prove to be a compelling motivation for the utility of differential equations. Furthermore, the growing complexity of the models leads to the development of various types of separable equations. Finally, survey data of the student experience of modeling indicates a general appreciation for modeling tasks such as this and a positive indication of their educational value.

The effectiveness of the STEMDISLEARN module in improving students’ critical thinking skills in the differential equations course

The effectiveness of the STEMDISLEARN module in improving students’ critical thinking skills in the differential equations course

Resolviendo ecuaciones diferenciales ordinarias con Symbolic Math Toolbox™ (Matlab) y SymPy (Python)

This paper shows solutions of ordinary differential equations (EDOS) obtained by using two symbolic packages: Symbolic Math Toolbox™ (Matlab) and SymPy (Python). The basic instructions to obtain solutions of both packages are explained step by step, through a group of examples from a traditional ordinary differential equations course. Differential equations that are solved with methods such as: separable variables, linear equations, indeterminate coefficients, variation of parameters, power series, Laplace transform, and numerical solutions are included. By means of the symbolic computation carried out with these packages it is possible to obtain the solution of linear systems, as well as the visualization of the direction field of a differential equation or of a non-linear system of differential equations. The main contribution of this work consists in providing the reader with a practical guide that allows him to start the study of differential equations assisted by Symbolic Math Toolbox™ or SymPy. Among the benefits of using these computational tools in teaching and/or learning practices, it is shown how the use of symbolic or numerical computation saves us effort in the computation of tedious calculations; focusing attention on important ideas and concepts such as: the relationship between the mathematical model and its physical counterpart, asymptotic behavior and qualitative analysis of the solutions.

RELATIONSHIP BETWEEN STUDENT DIFFERENTIAL AND INTEGRAL CALCULUS LEARNING OUTCOMES TO STUDENT DIFFERENTIAL EQUATIONS LEARNING OUTCOMES

Differential calculus and integral calculus courses are the foundations of mathematics that are important in understanding and applying differential equations. Students can take the Differential Equation course on condition that they have graduated from the Differential Calculus and Integral Calculus courses. This study aims to determine the relationship between the understanding of differential and integral calculus with students' learning outcomes of differential equations. The population in this study were all 2020 Pattimura University Teaching and Teacher Training Faculty Mathematics Education students who had taken differential calculus, integral calculus, and differential equations courses with a total sample of 33 people. The research method used is quantitative research with correlational methods. This study uses multiple regression analysis which has the aim of finding the form and influence between the independent variables and the dependent variable. Data analysis technique using multiple regression test. Based on the results of the data analysis, it was concluded that there was a significant contribution between learning outcomes of differential calculus on learning outcomes of differential equations seen in the relative contribution of learning outcomes of differential calculus on learning outcomes of differential equations based on the calculation results obtained 56.45% while the relative contribution of learning outcomes of integral calculus 43 .54% of the learning outcomes of differential equations. So it shows that there is a big influence between learning outcomes of differential calculus and integral calculus on learning outcomes of differential equation.

Variational study of the pendulum’s angular motion in a differential equations course

The objective of this research is to get students to mean the mathematical objects related to the second order differential equation through the context of a physical phenomenon. With the support of the physical system of the simple pendulum, the behavior of the angular position is studied in a modelling process with the help of simulators. From variational thought and language as a line of research of socio-epistemology in mathematics education, activities that promote a modelling process in a differential equations course for students in initial training for teaching mathematics are designed, implemented, and analyzed. The results of the process experienced by the students are presented, such as: familiarization of the phenomenon, use and interpretation of the simulations, construction of the mathematical model and its predictive role for decision making. It was identified that the students’ experiences, as well as the physical context, were determining elements for success in the resolution of the design.

Teaching and Learning of Vertex Coloring Using an Inquiry-Based Approach

Traditional lecturing is the dominant method of teaching mathematics in undergraduate mathematical courses in many countries, whereas active student-centered approaches such as inquiry-based learning have been shown in some instances to be more effective. Most teaching resources (inquiry-based tasks) available at the tertiary level are related to calculus, linear algebra, and differential equations courses. In this paper, we focus on using an inquiry-based approach to explore the teaching and learning of vertex coloring, a common topic in graph theory and undergraduate discrete mathematical courses. We designed seven inquiry-based tasks related to vertex coloring to teach vertex coloring in an undergraduate graph theory course to students majoring in mathematics. In sharing these tasks, describing students’ engagement with them, and discussing how these tasks were perceived by the students and lecturer, we hope to promote using inquiry-based tasks in undergraduate discrete mathematical courses.