1
vote
0answers
44 views

Insightful books on differential equations?

What are some recommendations for insightful books on differential equations and difference equations? These books don't need to be in the format of a textbook, and don't need to provide the same ...
0
votes
2answers
23 views

What are the complex solutions of a linear homogenous ODE of order $n$ with constant coefficients?

What are the complex solutions of a linear homogenous ODE of order $n$ with constant coefficients? Where can I read a proof? p.s. I don't even see the answer to the first question with a google ...
1
vote
1answer
47 views

“Reparametrizing” a differential system of the first order (Vinograd theorem?)

Consider a continuous function $f:\Omega\subset\mathbb R^n\longrightarrow \mathbb R^n$ such that for every $x\in\Omega$ the Cauchy problem: $$(\ast)\left\{\begin{array} {ll} y'=f(y)\\ y(0)=x ...
1
vote
0answers
21 views

System of ODEs and DAE system

Let us consider the following system of ODEs: $$ y' = f(y,z),\quad z' = g(y,z),\quad y(0) = y_0,\;z(0)=z_0 $$ and the following one: $$ y' = f(y,z),\quad 0 = g(y,z), \quad y(0) = y_0. $$ $f$ and $g$ ...
4
votes
0answers
97 views

Operators such that $\langle Ax,x \rangle=-\langle x,Ax \rangle$

Let $X$ be a Banach space. We consider the differential equation: $$x'(t)=Ax(t), \ \ \ t\in\mathbb{R}$$ where $A$ is a bounded operator on $X$. If $X$ is a Hilbert space, and $x(t)$ is a solution of ...
0
votes
1answer
45 views

Positive operators in Hilbert spaces

Let $H$ be a Hilbert space. I am just asking if there's some reference which studies operators $A$ with this property: $$\left\langle Ax,x\right\rangle \geq0,$$ for all $x\in H$. And $Ax=0$ whenever ...
2
votes
0answers
24 views

List of IVP known to have periodic solutions

I am looking for a list or review article describing differential equations and corresponding initial conditions which result in periodic solutions.
1
vote
0answers
25 views

Application of operator theory in ODE and PDE

I am looking for references of applications of operator theory (especially spectral theory) in ODE, PDE and possibly SDE. I have learnt operator theory in the general set up, but only know little ...
1
vote
0answers
40 views

Best way of learning dealing with DEs?

I'm from a cognitive science/computer science background. As I'm currently dealing with neuroscience a lot, I want to get better at dealing with DEs. My primary goal is to learn the tools required for ...
2
votes
1answer
32 views

Uniqueness of the ODE solutions

Say we have a continuous function (perhaps not everywhere differentiable) that satisfies an ODE $y^\prime(x)=h(y(x),x)$ for almost all $x$ in $[0,1]$. Are the any references for that deal with basic ...
1
vote
2answers
58 views

Need help on books on diff. equations/geometry and theoretical computer science

I am looking for recommendation of 3 different books on the following topics: 1.Differential Equations -Ordinary diff. equations -Vector field, transport equations -Equation of wave and heat -Use ...
4
votes
4answers
137 views

A question regarding Frobenious method in ODE

Suppose $b(x),c(x)$ are real functions analytic at 0. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. Suppose $r$ is a double root of $r(r-1)+b_0r+c_0=0$. It is well ...
0
votes
2answers
19 views

Pendulum reference request

I found on the net the following excerpt Could someone help me giving the title, author(s) of this book?
3
votes
1answer
66 views

Differential equations books using lots of algebraic topology?

The wikipedia page on 'Algebraic Topology' contains the following sentence: One can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf cohomology to ...
7
votes
0answers
98 views

Reference request: a differential equation arising in geometry

$$ \frac{d\beta}{d\alpha} = \frac {\sin\beta}{\sin\alpha} $$ In what contexts (if any) is this equation known to occur?
0
votes
1answer
57 views

Reference request: Gronwall's inequality with negative sign(s)

The following claim is a consequence of Gronwall's theorem Let $x \colon [0,\infty) \to \mathbb R$ with $x(0) = 0$ be a continuously differentiable function, whose derivative satisfies $$ \dot ...
0
votes
0answers
13 views

what does well posdeness results tells us concerning non linear evolution equations?

Consider a nonlinear Shr\"odinger equation, $$iu_{t}+\bigtriangleup u + f(u)= 0, u(0)= u_{0}$$ where $u(t, x)$ is complex valued function of $(t,x) \in \mathbb R \times \mathbb R^{n}$, $i=\sqrt{-1}, ...
0
votes
1answer
49 views

Recommend resources on dynamical systems and singularities

I'm looking for resources on bifurcation theory and systems of non-linear differential equations, but am very particular about the way it is taught/explained. I would like the approach to be based on ...
2
votes
1answer
51 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 ...
2
votes
1answer
69 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
303 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
80 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
73 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
69 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
32 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
160 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
40 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
160 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
1answer
60 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
85 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
52 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 ...
2
votes
4answers
161 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
177 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
116 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
47 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
1answer
43 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
76 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
107 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
116 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
35 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
72 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
189 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
57 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
155 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
73 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
61 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
92 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 $f\in C_c^\infty(0,T)$, $u\in L^1(0,T)$ ...