2
votes
1answer
23 views

Reference Request - Series Solutions to Differential Equations

I am looking for a text that gives a good exposition of power series solutions to second order equations with variable coefficients. My course I'm guessing focuses mainly on this section. My knowledge ...
0
votes
0answers
41 views

What would be a good source to learn differential equations?

I have done my school mathematics well but I have had difficulties in modeling even relative easy problems to differential equations. Are there some good books or some other material to learn to ...
2
votes
1answer
64 views

Literature for ODE undergrad class

I am teaching a undergrad ODE class. I am looking for some good (introductory) articles with applications of ODE's. In particular I would like some motivations for some special functions (Legendre, ...
0
votes
0answers
32 views

operator onto-theorem

I have this theorem: Let $V$ a Banach space, reflexive,separable, and let $A$ an operator monotonic, bounded, semi-continuos, coercive. Then, $A$ is onto. Where we can find the proof of this ...
5
votes
2answers
269 views

A proof of a theorem of Liouville

I need some reference for the proof of the following theorem attributed to Liouville: Theorem: Let $f(x):\Omega\longrightarrow \mathbb R^n$ a $C^2$ function where $\Omega$ is an open subset of ...
2
votes
1answer
56 views

What ODE book has good exercises?

What book has good exercises for ODE? I would say I am just starting to study the subject rigorously, but I am pretty well-versed at math more broadly. I am reading the intro-level book by Coddington, ...
1
vote
1answer
70 views

Solve the equation $a(x)y' + b(x)y=c(x)$. What if $a(x)=0$ somewhere? Does it always blow up?

Solve the equation $$a(x)y' + b(x)y=c(x)$$ (Hint: integrating factor.) I believe that if we assume $a(x)$ is differentiable and nonzero on an interval $(\alpha, \beta)$, we can use the ...
0
votes
2answers
46 views

Resources for teaching introductory course in differential equations?

The first time I was assigned to teach an introductory linear algebra course, I was able to find a number of resources which were helpful. For example, Linear Algebra Gems and Resources for Teaching ...
0
votes
0answers
23 views

Smooth paths and homotopies

In applications of the fundamental group(oid) to smooth manifolds it is sometimes useful to have paths which are smooth, rather than merely continuous. For example, if we consider the local system of ...
2
votes
2answers
92 views

Reference book for “Dynamical Systems”

I want to do my thesis about oscillations. I am a math student so I enjoy rigorous texts and hate sketchy ones. I am looking for a textbook or a good source that could help me with dynamical systems. ...
1
vote
2answers
31 views

I need a reference for: Existence and Uniqueness of a general ODEs with a linear operator

I'm looking for a reference of a theorem that establishes the existence and uniqueness of the following general ODE: Let $Q_n$ is a finite dimentional Hilbert space and let the operator $A:Q_n\to ...
2
votes
2answers
35 views

Reference Request for Linear ODEs

Homogeneous, linear ODEs of the form $$\mathrm{f}^{(n)}(x)+a_{n-1}\mathrm{f}^{(n-1)}(x)+\cdots+a_1\mathrm{f}'(x)+a_0\mathrm{f}(x)=0$$ where each $a_i \in \mathbb{R}$ are known to have "solution ...
11
votes
2answers
145 views

Topological equivalence of ODEs

Let's have ode $x' = f(x)$, $f(0) = 0$, $x\in \mathbb{R^n}$. There is clasical theorem that states that if all eigenvalues of ${\rm Df}(0)$ have nonzero real part, than $x'=f(x)$ and $x'={\rm Df}(0) ...
2
votes
0answers
44 views

Reducing size of ODE system by using symmetries: examples, references help request.

We know: A high order differential equation can be expressed as an ODE system. Knowledge of a symmetry allow one to reduce the order of a differential equation. So if we do $n$-order ODE ...
1
vote
1answer
67 views

Good ODE Books That Explain How Solution Methods Came To Be and Their Justifications

As part of the mathematics program offered at my college, I took an introductory ODE course a few semesters back. This was the one math course in my entire college career that I was totally lost in. ...
2
votes
1answer
47 views

ODE system and single PDE “equivalence”, reference request

The answers to this question Replacing large-dimensional ODE systems with one PDE suggest that, in general, one can not hope for "replacing" an ODE system with a single PDE. On the other hand, this ...
1
vote
4answers
128 views

Pde book suggestion.

I am studying PDE. And I want to know introduction to PDE book's names, which contain direclet problem, Sturm liouville problems, cauchy problems, euler, eigen functions and like this. But the ...
7
votes
2answers
156 views

Wave-Particle Duality in PDE?

I am reading Arnold's Lectures on Partial Differential Equations. It is definitely a good book, yet sometimes I am a little bit confused. One theme of the first chapter seems to be From the ...
2
votes
2answers
86 views

Ideas about an Ordinary Differential Equations research work (University level)

Good afternoon to everyone, I need some ideas about a Ordinary Differential Equations research work. It is for the ODE subject that I am doing at my Mathematics degree in my University. They asked me ...
3
votes
1answer
64 views

$\dot y=y^2-t$ Differential Equation

Liouville proved that the differential equation $\dot y=y^2-t$ does not have a solution in form of algebraic equations. Do you know any reference where I can find the proof for that?
1
vote
1answer
39 views

References about non uniqueness of solution in ordinary differential equations

I am looking for some example of IVP with non unique solution. I already know the examples of $f(x,y)=k\sqrt{y}$ and $f(x,y)=k y^{2/3}$. Any book or link is welcome.
0
votes
0answers
30 views

Existence of solutions to linear evolution equation with a noncoercive operator

Consider the Gelfand triple $V\hookrightarrow H \hookrightarrow V'$ and, for given $T>0$, the Sobolev-Bochner space $$ \mathcal W(0,T) := \{ v \in L^2(0,T;V): \dot v \in L^2(0,T;V')\}. $$ Consider ...
1
vote
2answers
69 views

Book searching in Elliptic Equation

I am learning a course with the subject of Elliptic Equations. If you know about it, please recommend me a book on Elliptic Equations. And if that's possible, someone post these books/author/...that ...
1
vote
0answers
94 views

Monodromy Groups of Differential Equations

I have heard that monodromy groups and analytic continuation can be used to construct new solutions to a differential equation from a particular solution. What references (textbook, or papers) could I ...
0
votes
1answer
99 views

good textbook to self-learn systems of ODEs

I've taken regular Ordinary Differential Equations. Right now I'm taking Systems of ODEs and the textbook is less than stellar. I was wondering if anyone could point me to a decent self-study book for ...
1
vote
1answer
32 views

German Books in Qualitative ODE?

Can anyone refer to me some good german books on introductory, qualitative ODE that cover topics like Picard-Lindelof, Peano's Existence Theorem?
1
vote
0answers
65 views

Is there a generalization of the ODE Comparison Theorem to n dimensional systems such as this one?

Is the following theorem true? If so, under what conditions? If not, why not? For any finite set of points $S$, let $conv(S)$ denote the convex hull of $S$. Let $f:\mathbb{R}^{n+1} \to \mathbb{R}^n ...
3
votes
1answer
177 views

Exercises about Distributions

I'm looking for references (books or pdf) about the following themes (especially the first two) : Fourier Series of Distributions. Distributional solutions of ordinary differential equations. ...
1
vote
0answers
56 views

A method called “incorrect method”

Good night. Is there a method called "incorrect method" to calculate second order differential equations? If so, please, is there a web page about it, as I have to investigate this method? Thank ...
4
votes
2answers
149 views

Dynamical systems and differential equations reviews/surveys?

I would be very glad if someone could point me to modern reviews/surveys on these topics. To be concrete, I'll provide some examples: S. Smale, Differentiable dynamical systems D. V. Anosov, On the ...
2
votes
1answer
65 views

Inverse problem in calculus of variations

I am interested in knowing which differential equations follow from a variational principle. I am reading this and it provides the answer for ordinary differential equations. Is there a complete ...
2
votes
1answer
57 views

Logistic model differential equation

I need to find the solution of the IVP $$\frac{dp}{dt}=ap-bp^2, p(0)=p_0$$ where $a,b$ are constants. I have found $$p(t)=\frac{ap_0}{bp_0+(a-bp_0)e^{-at}}$$ Now if $p(t_1)=p_1$ and $p(t_2)=p_2$ ...
0
votes
1answer
78 views

Reference Request: Weak ODEs and weak Gronwall inequality

During my research I came across a weak gronwall-type inequality of the following type: $$-\int_0^T f'(t)(u(t)-u_0) \leq \int_0^T f(t)u(t)$$ for non-negative $f\in C_c^\infty(0,T)$, $u\in L^1(0,T)$ ...
2
votes
4answers
246 views

A book 0n ODE ..

Please i want to request for recommendation of a book that can be used to have a good grasp of the concepts of ODE. Mainly the uniqueness, existence, stability and various proofs and applications of ...
2
votes
2answers
108 views

Where can I learn to apply the *theory* of differential equations in order to solve problems rigorously?

I'm having trouble solving ordinary differential equations (DE's), because I don't understand the underlying theory, nor how to apply it. For example, suppose I am solving such a DE. My techniques ...
0
votes
4answers
97 views

Book Searching in Stability Theory.

Can anyone recommend me a book on Stability Theory with an intuitive approach? I have some course notes on that subject, but it's really abstract and theoretical. I really want to understand it, ex: ...
6
votes
2answers
209 views

How to interpret the meaning of “$y$ solves the DE” to have nice properties.

Assume that $I$ is an open interval $0 \in I$ $x$ varies in $I$ $y$ is a differentiable function of $x$. Now in the context of these assumptions, consider the following problem. ...
2
votes
0answers
44 views

A system of ODEs, what existence results are there?

Let $u(t) \in \mathbb{R}^n$. Are there existence results for the ODE $$C(t)u'(t) = A(t)u(t) + f(t)$$ where $A(t), C(t) \in L^\infty(0,T;\mathbb{R}^{n\times n})$, $f(t) \in L^2(0,T;\mathbb{R}^n).$ In ...
3
votes
2answers
66 views

what is name of this numerical scheme for ode?

Let's have system of ODEs $$ \dot x(t) = A(t)x(t) $$ I came up with this numerical scheme: $$ x_{n+1} = e^{\frac{h}{2}A(t_{n+1})}e^{\frac{h}{2}A(t_n)}x_n $$ where $h$ is time step, $t_n = nh$ and ...
4
votes
1answer
541 views

Finding Weak Solutions to ODEs

I'm wondering if anyone has a reference to a good set of notes on finding weak (distributional) solutions to ODEs, or has any tips or tricks. For example, $$ xy^\prime=0 $$ has a classical solution ...
2
votes
1answer
343 views

How to solve differential equations using fft?

Can anyone point me to the principles and books/websites about it? Which properties must the differential equation have that a solution with fft is possible? Why can it be solved that way?
5
votes
1answer
96 views

Matrix differential equation and closed orbits

everyone. I am asking for a reference for the nonexistence of closed orbits (periodic orbits) of Matrix differential equations of the form \begin{equation} v\prime = M(v)\cdot v \end{equation} where ...
3
votes
4answers
184 views

Textbook Recommendation: Topological Dynamics

I need to take credits satisfying a topology requirement, and can structure it myself. My field of study is dynamical systems, can someone recommend a textbook that handles differential ...
2
votes
1answer
1k views

Numerically solving a system of nonlinear ODEs with boundary conditions

I have a system of 6 second-order nonlinear ODEs involving 5 different functions of a variable $t$. Every function has a boundary condition at $0$. I've never taken a differential equations class and ...
1
vote
2answers
121 views

Understanding differentials

What is a good reference to learn about differentials and related topics. Some of my questions are: Why is it possible to split $dy/dx$ into individual terms $dx$ and $dy$? In a separated ...
1
vote
2answers
98 views

Approximation of differential equations

Can someone provide me a good reference about approximation techniques in continous domain (not piecewise nor numerical methods) for differential equations?
11
votes
0answers
219 views

On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
5
votes
1answer
150 views

Delay-differential equation

Consider the equation $$ f'(t)=\frac{f(t-b)}{t-b}$$ $f'(t)=\frac{df(t)}{dt}$ and $b$ is a constant. Does anyone know if this equation has a name, an analytic solution and how to find the solution? ...
3
votes
4answers
340 views

Differential Equations Reference Request

Currently I'm taking the Differential Equations course at college, however the problem is the book used. I'll try to make my point clear, but sorry if this question is silly or anything like that: the ...
2
votes
0answers
61 views

Looking for online matlab-based differential equations course/text.

I am looking for an online ODE course that would be matlab/project-oriented. A full online text/course in the spirit of this linear algebra text is preferred. I know about the following CODEE and ...