Elementary Differential Equations and Boundary Value Problems.

Bailey, D. F.; Boyce, William E.; DiPrima, R. C.; Braun, Martin

doi:10.2307/2321040

Cited by 108 publications

(174 citation statements)

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“…The teaching experiment was organised in a DEs course for senior mechatronics students in their fourth year of studies. A total of thirty-seven students enrolled in this course based on a popular textbook by Boyce and DiPrima [4]. In the final part of the course, students worked for three weeks on an assessed assignmenta set of non-routine problems on EUTs designed by the authors with the focus on the development of conceptual understanding.…”

Section: Teaching Experiments and Data Collectionmentioning

confidence: 99%

“…Since the nature of sixteen out of seventeen problems on EUTs in the course textbook [4] is procedural, the problems in our assignment were designed to engage students in a deeper reflection about EUTs and related notions. In this paper, we discuss students' approach to the following two problems.…”

Section: Sample Tasks and Expected Reasoningmentioning

confidence: 99%

“…In this paper, we discuss students' approach to the following two problems. Both problems are non-routine; no similar examples or problems are discussed in the textbook [4]. The tasks encourage exploration and set cognitive demands at the higher levels of using procedures with connections to concepts and meanings and doing mathematics [21].…”

Section: Sample Tasks and Expected Reasoningmentioning

confidence: 99%

“…B4. We have a linear equation, like 𝑥𝑥𝑦𝑦 ′ + 2𝑦𝑦 = 18𝑥𝑥 4 , and if we put it in the standard form, we have a coefficient like 𝑝𝑝(𝑥𝑥) = 2 𝑥𝑥 and that will be discontinuous at 𝑥𝑥 = 0. And according to the existence and uniqueness theorem does a solution satisfying the initial condition 𝑦𝑦(0) = 0 exist or not?…”

Section: Episode 2 -Problem 1 (C)mentioning

confidence: 99%

See 3 more Smart Citations

Promoting conceptual understanding of differential equations through inqiury tasks

Rogovchenko¹,

Rogovchenko²

2022

Towards a New Future in Engineering Education, New Scenarios That European Alliances of Tech Universities Open Up

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Courses in Differential Equations (DEs) have been an important part of engineering education for decades. However, students experience difficulties with the understanding of main concepts including differential equation itself and diverse types of solutions (general, particular, stationary). In this paper, we discuss how the work on non-routine problems on the Existence and Uniqueness Theorems (EUTs) helps students to make sense of DEs and their solutions thus contributing to the development of advanced mathematical thinking.

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Section: Teaching Experiments and Data Collectionmentioning

confidence: 99%

Section: Sample Tasks and Expected Reasoningmentioning

confidence: 99%

Section: Sample Tasks and Expected Reasoningmentioning

confidence: 99%

Section: Episode 2 -Problem 1 (C)mentioning

confidence: 99%

See 2 more Smart Citations

Promoting conceptual understanding of differential equations through inqiury tasks

Rogovchenko¹,

Rogovchenko²

2022

Towards a New Future in Engineering Education, New Scenarios That European Alliances of Tech Universities Open Up

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show abstract

“…Laplace transform has a special place in solving differential equations with initial conditions. In this transformation, the appropriate function f of the transform t is transformed into a function like F of the transform s, and the function is transferred from the space t to the space s. In fact, by using the Laplace transform, a linear differential equation (including a function of the transform t) with the initial conditions becomes an algebraic equation in terms of s becomes [2].…”

Section: Introductionmentioning

confidence: 99%

Application of Laplace Transform in Solving Linear Differential Equations with Constant Coefficients

Naimi¹,

Mohammadi²,

Anwari³

2023

Technium

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In recent years, the interest in using Laplace transforms as a useful method to solve certain types of differential equations and integral equations has grown significantly. In addition, the applications of Laplace transform are closely related to some important parts of pure mathematics. Laplace transform is one of the methods for solving differential equations. This method is especially useful for solving inhomogeneous differential equations with constant coefficients and it has advantages compared to other methods of solving differential equations. Linear differential equations with constant coefficients are among the equations that can be solved using the Laplace transform. Because the transformation Laplace is one of the transformations that easily converts exponential functions, trigonometric functions, and logarithmic functions into algebraic functions. Therefore, it is considered a better method for solving linear differential equations with constant coefficients.

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Electromagnetic Waves in Crystals: The Presence of Exceptional Points

Sturm

2024

Advanced Photonics Research

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Although the investigation of the propagation of electromagnetic waves in crystals dates back to the 19th century, the presence of singular optic axes in optically anisotropic materials has not been fully explored until now. Along such an axis, either a left or a right circular polarized wave can propagate without changing its polarization state. More generally, these singular optic axes belong to exceptional points (EPs) in the momentum space and correspond to a simultaneous degeneration of the eigenmodes and their propagation properties. Herein, a comprehensive discussion on EPs in optically anisotropic materials, their occurrence, and properties as well as the properties of the electromagnetic waves propagating along such EPs is presented. The presence of such EPs, their spatial and spectral distribution in bulk, and semi‐infinite and finite crystals are discussed. It is shown that the presence of interfaces has a strong impact on the presence of the EPs and their spatial distribution. At an EP, the propagation of an arbitrarily polarized wave cannot be described by a superposition of two eigenmodes, as typically described in textbooks. This leads to singularities if the reflection and transmission coefficients have to be calculated. Here, two approaches are presented to overcome these limitations.

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Elementary Differential Equations and Boundary Value Problems.

Cited by 108 publications

References 0 publications

Promoting conceptual understanding of differential equations through inqiury tasks

Promoting conceptual understanding of differential equations through inqiury tasks

Application of Laplace Transform in Solving Linear Differential Equations with Constant Coefficients

Electromagnetic Waves in Crystals: The Presence of Exceptional Points

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