Tagged Questions
2
votes
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 ...
1
vote
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 ...
0
votes
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.
$$\dot x=P(x,y)$$
$$\dot ...
0
votes
1answer
60 views
Results for $y^{\prime\prime}(x) = a(x)y(x)$, where $a(x) > 0$.
I'm looking for references to any known results regarding solutions to the following 2nd order ODE $y''(x)=a(x)y(x)$, where $a(x)>0$ and $x \in \mathbb{R}$. Any help would be appreciated.
1
vote
1answer
109 views
Does a bound on a solution to an ODE allow for it to be defined over all $t \in \Bbb R$?
Consider the ODE $$ x^{(n)}(t) = f(t, x, x^{(1)}, \dots, x^{(n-1)})$$
Much of the books I have read through talk about results for very loose conditions on $f$. My first question is are there any ...
0
votes
0answers
32 views
References that discuss systems of ODEs on the non-negative orthant of $\mathbb{R}^n$?
Does anyone know of any references discussing initial value problems on the non-negative orthant? More specifically, consider the initial value problem
$\frac{dx}{dt}=f(x),\quad\quad ...
0
votes
0answers
28 views
Are there references on solving a system of first order nonlinear odes?
I would like to learn about what kinds of systems of first order nonlinear odes may have exact solutions, while trying to solve my previous question (which is a system of nonlinear, separable and ...
1
vote
0answers
90 views
George Simmons' “Differential Equations with Applications and Historical Notes” vs. “Differential Equations: Theory, Technique, and Practice”
I've heard much acclaim for George F. Simmons' "Differential Equations with Applications and Historical Notes" (2nd edition). I've noticed there's a newer book by Simmons and Krantz entitled ...
1
vote
0answers
34 views
Solutions of linear ODE with quadratic coefficients (reference request)
I am interested in the linear differential equation:
$$ \dot{x}(t) = (A t^2 + Bt + C) x(t)$$
where $A, B, C \in \mathbb{R}_{n \times n}$ and $x(t) \in \mathbb{R}^n$. Does anyone know about the ...
3
votes
1answer
127 views
Looking for a logically coherent book for the self-study of differential equations
I'm looking for a logically coherent book for the self-study of differential equations. Let me clarify.
By logically coherent, I don't mean proofs of the limit laws, uniqueness theorems etc.
By ...
2
votes
3answers
203 views
Mathematical applications of ordinary differential equations.
I'm looking for more mathematically oriented applications of ODEs (if possible of first order equations). I've browsed through several books and they are all full of physics applications and very ...
3
votes
0answers
26 views
Links to pdf-articles or books where there is an information on some linear integral operator
Please write me links to pdf-articles or books where there is some information on properties of operators like these:
$$
(Af)(x,y)=\int_{D}\frac{f(z) \, dz}{|x-z| |z-y|}
$$
or
$$
(Bf)(x,y)=\int_D ...
1
vote
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 ...
0
votes
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 ...
1
vote
0answers
46 views
ODE: continuous dependence on parameters
Is it true that the solutions of the problem:
$$\begin{cases} \frac{\text{d}}{\text{d} s} [s^{2-2/N} u^\prime (s)] + \frac{\lambda}{c_N^2}\ u(s)=0 \\
u(\bar{s})=1\\ u^\prime ...
5
votes
6answers
390 views
Free differential equations textbook?
I've seen questions on what are some good differential equations textbook and people generally points to Ordinary Differential Equations by Morris Tenenbaum and Harry Pollard and so on
I was ...
14
votes
1answer
210 views
How does one parameterize the surface formed by a *real paper* Möbius strip?
Here is a picture of a Möbius strip, made out of some thick green paper:
I want to know either an explicit parametrization, or a description of a process to find the shape formed by this strip, as ...
0
votes
1answer
84 views
Harmonic Extension
Let be $u$ a harmonic function defined on an open set $\Omega \setminus \{p\} \subset \mathbb{C}$ of the complex plane. Show that if $u$ is bounded in a neighborhood of $p$ then $u$ admits a harmonic ...
2
votes
3answers
67 views
Is following system of nonlinear ODEs recognized?
The following system of ODEs – is it recognized as distinct system, with meaningful background and uses?
$$\frac{dx}{dt} = - [x(t)]^2 - x(t)y(t)$$
$$\frac{dy}{dt} = - [y(t)]^2 - x(t)y(t)$$
This is ...
2
votes
1answer
355 views
Looking for a book on Differential Equations *with solutions*
I'm studying differential equations (specifically Laplace Transforms) right now with my college assigned 'Differential Equations with Application and Historical Notes'-George F Simmons. While I like ...
7
votes
0answers
154 views
Addition formula for $f_n(x+y)$ in closed form.
$n$ is a positive integer.
$$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
...
2
votes
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 ...
0
votes
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 ...
6
votes
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 ...
4
votes
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 ...
3
votes
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 ...
0
votes
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 ...
4
votes
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.
1
vote
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)
$$
with starting positions $y(0) ...
0
votes
1answer
80 views
How to show existence and uniqueness of a SL problem with von Neumann BCs
Let $f\in C[0,1]$ be a continuous function and consider for $x\in(0,1)$ the Sturm-Liouvile problem $$ -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]$ ...
2
votes
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 ...
1
vote
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 ...
1
vote
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, ...
2
votes
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 ...
0
votes
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 ...
0
votes
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.
...
2
votes
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 ...
0
votes
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)}$ ...
1
vote
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
4
votes
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 ...
4
votes
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 ...
1
vote
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. ...
2
votes
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))$$
with a constrain of the kind:
...
3
votes
1answer
126 views
Can all first order ODEs be made exact?
Elementary differential equations classes usually cover exact differential equations. These are equations of the form: $$M(x,y)+N(x,y)y'=0 \qquad \mathrm{such\;that} \qquad \frac{\partial ...
2
votes
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 ...
1
vote
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?
0
votes
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 ...
5
votes
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!
2
votes
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 ...
2
votes
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 ...
