# Tagged Questions

1answer
46 views

### Problem books in ODE

I'm studying Ordinary differential equations right now in the level of Hartman's book. I've never seen problem books in ODE in this level even if you consider it without solutions. I would like to ...
2answers
69 views

### resources to study PDE from

I am an undergrad engineering student. I recently completed my second year, with that said, I have taken several calculus courses. Most recently I completed differential equations and multivariable ...
1answer
30 views

### Dynamical Systems. Bendixson's and Dulac-Bendixson's theorems.

I am looking for a place to read the proofs of Bendixsons and Dulac-Bendixsons theorems. Namely let D be a simply connected set and the following system be defined in D. $$\dot x=P(x,y)$$ \dot ...
1answer
60 views

### Results for $y^{\prime\prime}(x) = a(x)y(x)$, where $a(x) > 0$.

I'm looking for references to any known results regarding solutions to the following 2nd order ODE $y''(x)=a(x)y(x)$, where $a(x)>0$ and $x \in \mathbb{R}$. Any help would be appreciated.
1answer
109 views

### Does a bound on a solution to an ODE allow for it to be defined over all $t \in \Bbb R$?

Consider the ODE $$x^{(n)}(t) = f(t, x, x^{(1)}, \dots, x^{(n-1)})$$ Much of the books I have read through talk about results for very loose conditions on $f$. My first question is are there any ...
0answers
32 views

### References that discuss systems of ODEs on the non-negative orthant of $\mathbb{R}^n$?

Does anyone know of any references discussing initial value problems on the non-negative orthant? More specifically, consider the initial value problem \$\frac{dx}{dt}=f(x),\quad\quad ...
0answers
28 views

### Are there references on solving a system of first order nonlinear odes?

I would like to learn about what kinds of systems of first order nonlinear odes may have exact solutions, while trying to solve my previous question (which is a system of nonlinear, separable and ...
0answers
90 views

### George Simmons' “Differential Equations with Applications and Historical Notes” vs. “Differential Equations: Theory, Technique, and Practice”

I've heard much acclaim for George F. Simmons' "Differential Equations with Applications and Historical Notes" (2nd edition). I've noticed there's a newer book by Simmons and Krantz entitled ...
0answers
34 views

### Solutions of linear ODE with quadratic coefficients (reference request)

I am interested in the linear differential equation: $$\dot{x}(t) = (A t^2 + Bt + C) x(t)$$ where $A, B, C \in \mathbb{R}_{n \times n}$ and $x(t) \in \mathbb{R}^n$. Does anyone know about the ...
1answer
127 views

### Looking for a logically coherent book for the self-study of differential equations

I'm looking for a logically coherent book for the self-study of differential equations. Let me clarify. By logically coherent, I don't mean proofs of the limit laws, uniqueness theorems etc. By ...
3answers
203 views

### Mathematical applications of ordinary differential equations.

I'm looking for more mathematically oriented applications of ODEs (if possible of first order equations). I've browsed through several books and they are all full of physics applications and very ...
0answers
26 views

### Links to pdf-articles or books where there is an information on some linear integral operator

Please write me links to pdf-articles or books where there is some information on properties of operators like these: $$(Af)(x,y)=\int_{D}\frac{f(z) \, dz}{|x-z| |z-y|}$$ or  (Bf)(x,y)=\int_D ...
2answers
63 views

### Can someone recommend a good textbook for a 3rd year ordinary differential equations class?

The class that I'm taking is called, Intermediate Ordinary Differential Equations and our required textbook is called Differential equations, Dynamical Systems and an Introduction to Chaos by Morris ...
0answers
10 views

### Qualifying Parameters

you have two parameters, 1) rates of trees per land size, ranging from 30%-100%, and 2) rates of birds per land size, ranging from 5%-30% goal is that you're trying to find out which is overall ...
0answers
46 views

### ODE: continuous dependence on parameters

Is it true that the solutions of the problem: \begin{cases} \frac{\text{d}}{\text{d} s} [s^{2-2/N} u^\prime (s)] + \frac{\lambda}{c_N^2}\ u(s)=0 \\ u(\bar{s})=1\\ u^\prime ...
6answers
390 views

### Free differential equations textbook?

I've seen questions on what are some good differential equations textbook and people generally points to Ordinary Differential Equations by Morris Tenenbaum and Harry Pollard and so on I was ...
1answer
210 views

### How does one parameterize the surface formed by a *real paper* Möbius strip?

Here is a picture of a Möbius strip, made out of some thick green paper: I want to know either an explicit parametrization, or a description of a process to find the shape formed by this strip, as ...
1answer
84 views

### Harmonic Extension

Let be $u$ a harmonic function defined on an open set $\Omega \setminus \{p\} \subset \mathbb{C}$ of the complex plane. Show that if $u$ is bounded in a neighborhood of $p$ then $u$ admits a harmonic ...
3answers
67 views

### Is following system of nonlinear ODEs recognized?

The following system of ODEs – is it recognized as distinct system, with meaningful background and uses? $$\frac{dx}{dt} = - [x(t)]^2 - x(t)y(t)$$ $$\frac{dy}{dt} = - [y(t)]^2 - x(t)y(t)$$ This is ...
1answer
355 views

### Looking for a book on Differential Equations *with solutions*

I'm studying differential equations (specifically Laplace Transforms) right now with my college assigned 'Differential Equations with Application and Historical Notes'-George F Simmons. While I like ...
0answers
154 views

### Addition formula for $f_n(x+y)$ in closed form.

$n$ is a positive integer. $$f_n(x)^n+\left(\frac{df_n(x)}{dx}\right)^n=1$$ $f_n(0)=0$, $f_n'(0)=1$ then I am looking for the addition formula for $f_n(x+y)$ in closed form. if $n=1$ then ...
1answer
111 views

### What is a strong stable manifold?

For a dynamical system in $\mathbb{R}^n$ given by $\dot{x} = f(x)$, and a fixed point $p$, one defines stable and unstable manifolds at the point $p$. These are well documented, and a quick ...
0answers
88 views

### Differential Equations, Probability/Statistics, Optimization Problem - Relations?

While I am working on some physical/mathematical problems, I feel strongly that these three areas are almost the identical thing, except that they have different methods/from different aspects to ...
0answers
75 views

### Measure-driven differential equations

Background: I need some help to understand the concept behind measure-driven differential equations. The solution of an ordinary differential equation is continuous. In order to describe discontinuous ...
2answers
68 views

### What do I need to know to simulate many particles, waves, or fluids?

I've never had a numerical analysis course so I don't know what I need to know. I'm just wondering what kind of books I should get to make me able to simulate these things. I'm wanting to simulate ...
2answers
86 views

### Reference request: stability theory in infinite dimensional dynamical systems/ partial differential equations

I am looking for some references (text books, elementary review papers, journal articles etc) regarding the phenomenon of breakdown in stability for (nonlinear) partial differential equations, i.e if ...
2answers
360 views

### ODE book recommendation

I have just completed my first year study and know elementary analysis and a little bit functional analysis. I found that most of the ODE books just focus on calculation but no substantial explanation ...
2answers
126 views

### Method to solve $xx'-x=f(t)$

I would like to resolve this differential equation: $xx'-x=f(t)$ any suggestions (or any online texts on similar differential equation) please? Thanks.
1answer
60 views

### Comparison theorem for systems of ODE

Let vector-function $x(t)$ satisfy a differential equation $$\dot x = f(x),$$ and a vector-function $y($t) satisfy a differential inequality $$\dot y \leq f(y)$$ with starting positions \$y(0) ...
1answer
80 views

### How to show existence and uniqueness of a SL problem with von Neumann BCs

Let $f\in C[0,1]$ be a continuous function and consider for $x\in(0,1)$ the Sturm-Liouvile problem $$-u''(x)+x\cdot u(x)=f(x) \tag1$$ where $u'(0)=u'(1)=0.$ I need to show that for any $f\in C[0,1]$ ...
1answer
87 views

### Citable Reference for Picard's Theorem in Banach Space

I was wondering if anyone knew of a legitimate citable reference where Picard's Theorem on the existence of solutions to ODEs in Banach space is proven? For some reason I can only find proofs for the ...
0answers
77 views

### Literature on Riccati equations (algebraic and differential)

Advise me please some book on algebraic and differential Riccati equations: I'm interested in such questions as theorems of existence, uniqueness and extendibility of solutions of differential ...
1answer
69 views

### Exponential stability of inhomogeneous linear ODE's

Can anybody give me a good reference which under suitable assumptions discusses exponential stability of $0$ for the equation $\dot{u}_t = A(t)u(t) + b(t)$ Here $u_t\in\mathbb R^n$ is the unknown, ...
2answers
137 views

### Newtonian potential of a rotationally-invariant function

Lately I read up in the wikipedia article about the Newtonian potential, that for any compactly supported continuous function $f: \mathbb{R}^d \rightarrow \mathbb{R}$ that is rotationally invariant ...
0answers
64 views

### test function to check region of stability - A stability

I have an ODE that increments exponentially and need to 'use the test function method' to describe its stability region and whether it's A-stable. Can anyone point me to a resource that's written for ...
1answer
266 views

### References to re-learn differentiation and integration

I'm looking to re-learn "differentiation and integration", it has really been a long time since I touched the subject. I'm considering starting with Algebra then differentiation and integration. ...
3answers
100 views

### Concise ODEs reference?

Is there any text that I can use as a short reference for the standard techniques for solving basic ODEs? I currently have been using Boyce and diPrima as my ODEs text, and it is far too wordy for my ...
1answer
88 views

### Basic Reference material about ODEs such as saparability with calculations and examples?

I am trying to show this kind of non-linear $y''''=y'y''/(1+x)$ in normal form. For example here if $y=e^{x}\rightarrow y^{(n)}=e^{x}\rightarrow x=-1$, where $y^{(n)}$ ...
1answer
133 views

### Suggestions for a Global Analysis book

can somebody tell me some good books or lecture notes in "global analysis" ? I am a newcomer in this subject. thanks in advance. greetings trito
3answers
285 views

### Theory of the Mathieu Operator

How important is the theory of the Mathieu operator in mathematics/applied mathematics? What are the major mathematical concepts required to study it? The Mathieu operator is an ordinary periodic ...
2answers
293 views

### Essay about the art and applications of differential equations?

I teach a high school calculus class. We've worked through the standard derivatives material, and I incorporated a discussion of antiderivatives throughout. I've introduced solving "area under a ...
3answers
1k views

### What is a good differential equations textbook?

I have taken a lot of math in university, but chose to omit differential equations. Unfortunately, now I have to read computer science proofs that use them, mostly ODEs, and this is always a struggle. ...
1answer
343 views

### Numerical solving a constrained system of differential equation

I am in trouble on finding a numerical technique to solve the following system of equations $$\ddot q_1(t)=f_1(q_1(t),q_2(t))$$ $$\ddot q_2(t)=f_2(q_1(t),q_2(t))$$ with a constrain of the kind: ...
1answer
126 views

### Can all first order ODEs be made exact?

Elementary differential equations classes usually cover exact differential equations. These are equations of the form: M(x,y)+N(x,y)y'=0 \qquad \mathrm{such\;that} \qquad \frac{\partial ...
2answers
278 views

### Differential Equations reference for Putnam preparation

I have two problem collections I am currently working through, the "Berkeley Problems in Mathematics" book, and the first of the three volumes of Putnam problems compiled by the MAA. These both ...
2answers
93 views

### High order methods for solving ODEs

I would like to know about really high order methods for solving ODEs. Say of order 30 and higher. What are they? Any surveys/reviews?
1answer
82 views

### Book on stability theory

I am looking for a book on stability theory. More precisely, I am interested in the case of a system of differential equations $\frac{dx}{dt}=Ax + F(x),$ where $A$ is a constant matrix, such that two ...
4answers
154 views

### Where can I find good, free resources on differential equations?

I'd like to know if there are any good online books, lecture notes, videos, tutorials, or similar that are free to the public (on differential equations). Suggestions are welcome!
3answers
199 views

### Numerical Analysis References

Could anyone suggest any good (perhaps online ref papers) reference material on numerical analysis focusing on determining accuracy/estimated errors, rates/orders of convergence especially when ...
0answers
233 views

### Which branch of mathematics is this and what are the introductory references?

I am self-studying a physics textbook on waves. While discussing solutions to linear homogeneous ODEs, the author talked about the exponential as "irreducible" solutions and on a footnote, said that ...
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 46, "mathjax_display_tex": 12, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9255381226539612, "perplexity_flag": "middle"}