Questions tagged [differential-equations]

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3 answers
198 views

A good way to learn differential equations rigorously

What would be a good books of learning differential equations for a student who likes to learn things rigorously and has a good background on analysis and topology?
4 votes
2 answers
1k views

Is the Wronskian still assumed for graduate education?

About thirty years ago, in a practice GRE (Graduate Record Exam) math test in the US, a question assumed the student knew the definition of the Wronskian. I had never heard of this determinant before. ...
3 votes
2 answers
83 views

Program for visualizing trajectories for a 2D system of linear differential equations

I am looking for a program so that when you give it a 2D system of linear differential equations and an initial condition, it can show an animation of the trajectory of a particle starting at that ...
4 votes
2 answers
160 views

Good Examples of Equations Derived from Elementary Calculus

I'm collecting additional enrichment content for my calculus students. I'm looking for examples of equations that are used in various fields, but which can be derived at least somewhat ...
0 votes
3 answers
298 views

How can I visualize differential equations and Integration in real life?

How can we understand differential equations and Integration in real life so that we can understand calculus easily. All we do here, at university level is memorize calculus and get the answer. We ...
3 votes
1 answer
278 views

What is the preferred method of teaching "linear" differential equations today?

For the purposes of this question, there are two kinds of differential equations: linear, and non-linear, which is to say that there could be two ways to teach the subject. One way is to separate ...
3 votes
3 answers
3k views

Are differential equations considered calculus and included in a calculus class or is it its own class?

Are differential equations considered calculus and included in a calculus class or is it its own class? Also, if it is its own class then what calculus classes does it come after?
5 votes
1 answer
137 views

Do people usually teach solving a linear differential equation by inverse operators in an undergraduate course?

For the following linear differential equation $$a_ny^{(n)}+\cdots+a_1y'+a_0y=Q(x),$$ most books teach the method of undetermined coefficients, variation of parameters and Laplace transforms. ...
4 votes
2 answers
342 views

Why are "homogeneous differential equations" in the standard ODE curriculum?

Here I mean a differential equation of the form $y'=f(x,y)$ where for some $\alpha$, we have $f(tx,ty)=t^\alpha f(x,y)$ for every $t$. I have no idea why this topic seems to appear in every ODE ...
3 votes
1 answer
132 views

Refreshing math knowledge

How do I refresh advanced math I learned at a graduate level? I once was able to do the full solution of a particle in a parabolic well and other advanced math, however 20 years later I'm struggling ...
9 votes
3 answers
2k views

Take-Home Examination on Ordinary Differential Equations?

I am planning to give my students a take-home examination on ODE. The main topic that I would like to cover is Linear Differential Equations of Order Greater than One. For example, I will give my ...
4 votes
1 answer
93 views

EdX Courses for Self-Study

I have been independently considering two edX courses in mathematics. The first, a course on probability theory drawing from a financial crisis case study, appeared to me plausibly comparable in ...
3 votes
0 answers
128 views

The propagation of the wave equation in even versus odd dimension

I am about to teach a second year undergraduate class on applied differential equation (first time) and, while I won't have time to go into the details, I wanted to show my students the difference ...
3 votes
1 answer
114 views

Applied ODEs for Numerical Methods

I am looking for a list of ODEs to use as examples in the teaching of a numerical methods course for engineers. I am looking for first and second order examples - the more applied (to engineering) ...
11 votes
1 answer
4k views

Diagram of Methods to Solve Differential Equations

I am currently trying to build a flow chart to visualize all tests there are to tell whether an ordinary differential equation is solvable and how to solve it. This is for tutoring purposes. The ...
11 votes
9 answers
10k views

Why do we study ordinary differential equations?

What is a good answer to the question: Why should one study ordinary differential equations? I would give the answer: ODEs are used in many models to determine how the state of this model is changing ...
4 votes
2 answers
147 views

How to explain linear approximation to an equation to calculus students?

I am, at the moment, teaching calculus to students whose majors are, for example, biology, biochemistry, chemistry and geology. The course book is Claudia Neuhauser's "Calculus for biology and ...
14 votes
3 answers
478 views

Theme of a Differential Equations Course

I am tasked with teaching an introductory differential equations course. In my experience, this course tends to be a laundry list of methods that apply to specific forms of differential equations ...
6 votes
1 answer
492 views

Why aren't integral equations often taught "back to back" with differential equations?

In single variable calculus, integral calculus is taught "back to back" with differential calculus. This is generally true, although to a lesser extent, with multivariable calculus. Yet I don't see ...
16 votes
4 answers
1k views

What is the motivation for characterizing second order linear PDEs as hyperbolic, elliptic, or parabolic?

I'm teaching an Intro to PDEs course (I'm an analyst, but PDEs are a bit outside my bailiwick) and I'm covering the basic examples: Heat, Wave, and Laplace. How should I move from these examples to ...
1 vote
1 answer
121 views

A robot to simulate differential equations for undergraduate students. [closed]

I was recently at EPFL drone days and enjoyed a demo of a robot that could follow a black line like in the sketch (I can improve the sketch on demand): Then I remembered my good all times at the ...
2 votes
3 answers
255 views

Easy tool to draw stencils

I'm looking for a simple tool (preferably online or at least that works on Linux) to draw stencils. When teaching and creating notes for students, I often feel that it would make understanding in some ...
6 votes
2 answers
389 views

A Plan for a Treatise Study of the Classical Theory of PDEs

The Plan In the study of any special issue in mathematics, two things may be of importance, namely, subjects and order of them. I just wrote down a plan to study the theory of partial differential ...
14 votes
1 answer
401 views

How to teach ordinary differential equations to good students?

I am TA-ing a introductory course on ODEs and PDEs this year. At my university most introductory math courses can be taken at "basic" and "extended" levels. This one is the extended one. My students ...
8 votes
2 answers
202 views

Second Order Differential Equation Example Request

I am looking for some non-complicated second order differential equations to illustrate certain techniques for control engineering. It doesn’t matter if the differential equations are linear or non-...
1 vote
3 answers
116 views

Workshop about ODEs and connected rates of change (A-levels)

I'm preparing a workshop about ODEs and connected rates of change for pre-college students (smart and interested in mathematics). I'd like to include some fun parts as well - even if these are ...
9 votes
2 answers
194 views

How much prior math should I review in teaching a graduate-level course?

I am scheduled to teach a graduate-level course in engineering whose basis is in the solution of ODE’s and PDE’s, and thus is about halfway between a math course and an engineering course. We ...
3 votes
2 answers
146 views

Monodromy examples for undergraduates

I'm looking to put together a 1 hour talk about monodromy aimed at undergraduate math and physics majors. In terms of prerequisites, I only want to assume the students have seen multivariable calculus,...
15 votes
4 answers
612 views

How would you explain what a PDE is to a very educated layman with no math background?

Is every mathematical concept, even the complex ones, explainable? As someone who will be needing to explain my line of work for a position to a committee who is very, very, educated, just not in ...
18 votes
3 answers
2k views

Should I teach Laplace Transforms? How much?

My question is in the title. Let me elaborate and give some context: I'm teaching a first differential equations course, essentially for engineers, at the university. I'm developing the syllabus ...
16 votes
4 answers
1k views

What is the right way to order the topics in a first ODEs course?

This question is long but I am asking for educated opinions on a question of math education and for this reason I'd like it not to be closed on the grounds that it invites subjective discussion. ...
6 votes
5 answers
259 views

Differential equations - definitions

I am having a great deal of trouble with the definitions used throughout the book so far - i.e. linear, homogenous, non-homogenous, etc. I am not sure why exactly they are useful to know. I am having ...
4 votes
0 answers
88 views

Split differential equations into two video courses; theory and calculation?

I'm creating a course on differential equations, and I'm painfully aware that there is a LOT to teach. As of right now, I'm considering splitting the course into two, in the following manner: theory ...
3 votes
2 answers
184 views

In what order should I teach methods for solving (linear) ODEs?

I'm teaching at high school level and the next topic will be differential equations. I would like my students to be able to solve some simple linear and non-linear ODEs using the ansatz $x(t) = e^{\...
6 votes
2 answers
498 views

Differential equations book

I am looking for a book on differential equations (ordinary/partial) of a particular kind. I figure out I am terrible at solving differential equations. This a vast field and unfortunately I don't ...
8 votes
3 answers
1k views

Commonly taught method divides by zero

I just saw this question and the two answers. Now I am very sure that if many teachers see these answers they would accept them. Unfortunately they are deeply flawed. The second one is clearly invalid ...
11 votes
1 answer
257 views

A more natural motivation for the appearance of generalized eigenvectors in linear system with repeated eigenvalue

When I teach constant coefficient linear differential equations, the usual guess of an exponential can be motivated because it is "approximately" a fixed point for the differentiation operator. The ...