Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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Different formulations of chemical kinetics giving different solution trajectories

I am reading a textbook by Keith J. Laidler titled 'Chemical Kinetics' (3rd ed.). Two different differential forms are given for the reaction (pp30, pp38): $ 2A \leftrightarrow B $ with forward rate $...
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44 views

Question about the assumption of a version of Grönwall's inequality.

According to Wikipedia, A version of Grönwall's inequality for the integral of continuous functions is the following: Let $I$ denote an interval of the real line of the form $[a,\infty)$ or $[a,b]$ ...
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127 views

Stability of Rossler System's fixed points - which methods to use?

linear stability analysis performed on the Rossler System yields the Jacobian $J=\begin{pmatrix} 0& -1&-1\\ 1 &a&0\\ z_0& 0&x_0-c \end{pmatrix}$ $z_0$, $x_0$ are the ...
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270 views

Contour Integral solution to differential equations, Euler transformation?

In Spain's book, Functions of mathematical physics he introduces the contour integral method of solving ODEs. The baseic idea is: given an ODE $\sum_0^m a_r(t) \frac {d^rf}{dt^r} = 0$, a solution may ...
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1answer
44 views

Linear equation and linear differential equations

I remember noting from an algebra class that $x$ and $y$ of a linear equation neither divide or multiply with each other which is somewhat clear from the forms of linear equations: General form of ...
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2answers
105 views

Getting 0 solving Schrodinger equation with Dirac delta by Fourier transform

I am attempting to solve the Schrödinger equation with the potential $V = - \delta (x)$. This leads to a differential equation $$ \alpha \psi''(x) + (E + \delta(x)) \psi(x) = 0 $$ where $$ \alpha \...
2
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1answer
59 views

Taylor Series General Formulas

I'm looking at 2 different Wikipedia pages: The formula here is different than the one given at the end of the section here. Aside from the remainder, why choose one over the other? I'm assuming ...
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61 views

Stability of an equilibrium

From a Center-Manifold reduction I get the following system: $$ \begin{pmatrix}\dot x \\\dot y\end{pmatrix}=\begin{pmatrix}-y(2x^2-2xy+y^2)\\x\end{pmatrix} $$ The aim is to analyze the stability of ...
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1answer
136 views

Kernel, Green function and the functional derivative.

I am pretty new to the subject of differential equations and am reading about Green functions and Kernels for the first time. I am more familiar with functional differentiation and am comfortable with ...
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63 views

In the Glycolysys Sel'kov model, what are the meaning of “a” and “b” values?

In the Sel'kov model of glycolysis which I put on next $u'=-u+av+u^2v\\ v'=b-av-u^2v$ which have a limit cycle and have all sense because it is a glycolytic cycle. What are the biological-medical-...
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82 views

Is it possible to bruteforce a differential equation

Is there any method to solve differential equations which involves just a number of basic functions combined into various permutations (with various factors) which are then fed into the differential ...
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41 views

Sturm-Liouville Problem: Possible range for the eigenvalues.

Let us consider the Sturm-Liouville Problem $$y''=\lambda \cdot y$$ with $y(0)=y(\pi)=0$. As only $\lambda \gt 0$ renders non-zero solutions, one obtains the condition $\sin(\mu \pi)=0$ for a non-...
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2answers
70 views

Difference in definition of differentiation

Okay so quite often I see two different definitions of differentiation and I want to know when it is appropriate to use each one. $$\lim_{h \to 0} \frac{f(x_{0}+h)-f(x_{0})}{h}$$ and $$\lim_{x \to ...
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57 views

Derivatives of solution to Schrödinger equation

Consider the differential equation (Schrödinger, but rewritten to be pleasing to Lie algebraic eyes): $\frac{d U(t)}{dt} = c(t)U(t)$ where $c(t)=a+w(t)b(t)$, $a,b \in \mathfrak{su}(n)$ and $w$ is a ...
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53 views

Does this type of differential equation have a name?

Does a differential equation of the form: $$y''(x)+\delta(x)y=Ay$$ where $\delta(x)$ is the Dirac Delta function and $A$ is a constant have a specific name?
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22 views

Step response for different definitions of step function

I was thinking about the solution of the known problem of determining the step response for the concentration leaving a CSTR tank. The differential equation is: $\frac{dC}{dt}=\frac{C_{0}-C}{\tau}-kC$...
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83 views

Do I have any hope with this PIDE?

$\frac{\omega(1-\omega)}{N_1} \frac{\partial^2 f}{\partial x_1^2} + \frac{\omega(1-\omega)}{N_2} \frac{\partial^2 f}{\partial x_2^2} + \cdots + \frac{\omega(1-\omega)}{N_k} \frac{\partial^2 f}{\...
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59 views

two point block method for solving ODE

How to solve the ordinary differential equation $$y'(t) = -1000 y(t)+ 999 e^{-t}, \hspace{10mm} 0≤t≤5.$$ $y(t)=e^{-t}$, for $t<0$. Using two point block method $$hf_{n+1}= \frac{1}{3} (hf_{n+2} -...
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46 views

Relationship between solutions of two matrix differential equations

Given a ($4\times4$ in the important case) matrix differential equation: $\frac{d U_t}{dt}= A_t U_t$ where $U_t \in SU(n)$ and $A_t \in \mathfrak{su}(n)$. What is the relationship between the ...
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38 views

Solution to system of ordinary differential equation

Given the system: $\begin{cases}x''=2y \\ y''=-2x\end{cases} $ I found the (I think) equivalent linear equation $x^{IV}+4x=0$ First question: is the equation actually equivalent to the system? ...
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29 views

Is there a physical meaning of ranking in differential algebra?

The main stone in the Ritt's Algorithm from differential algebra is ranking. If we consider an example of a differential polynomial with two variables $x$ and $y$. Then how can we say $x$ is ranked ...
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1answer
60 views

Recurrence equation approximation

I have the following recurrence relation, $$x_{i+1}=a\cdot x_i^{\frac{2-2\alpha}{3}}+x_i,$$ where $a>0, \alpha>0$, and $x_0>0$. My goal is to get an approximate the expression for $x_i$. I ...
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2answers
116 views

Physical applications of Chebyshev's equation.

As reported by Wikipedia, Chebyshev's equation is the second order linear differential equation $$(1-x^2) {d^2 y \over d x^2} - x {d y \over d x} + p^2 y = 0 $$ where $p$ is a real constant. Has ...
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31 views

Lowering the power of a linear differential equation.

$$L(x)\equiv x^{(n)}+a_1(t)x^{(n-1)}+...+a_{n-1}x'+a_n(t)=0.$$ The solutions $x_1, x_2,...,x_m (m<n)$ are given. Linearly independent. Let us find $x_{m+1},...,x_n$ It's starts off like this, I ...
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Limits of systems of differential equations.

Consider the following system of odes involving a real parameter $\epsilon$ as follows \begin{equation*} \frac{dx_i}{dt}=f_i(x) + \epsilon^2g_i(x,y) \ , \ \ \ \frac{dy_j}{dt}=h_j(y) + \epsilon^{-2}k_j(...
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1answer
129 views

Unique solution of Volterra integral equation of second kind

Dear Maths Stackexchange, In the context of a physics problem, I am looking at a linear integral equation, more specifically a 2nd kind Volterra equation in the unknown $g(t)$: \begin{equation} \...
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51 views

Extensions of $C^k$ functions to the boundary [closed]

Assume $\Omega \subset \mathbb R{^n} $ is an open connected smooth domain. I have some propositions that I guess they are correct , but I want to be confident. If $f\in C_0^k(\Omega) $ then $f\in C^...
2
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1answer
102 views

Simple Lotka-Volterra Slope Field in Phase Space

I'm trying to plot the slope field in phase space of a simple (all constants set equal to $1$) Lotka-Volterra system described by the following differential equations: $$\frac{dw}{dt} = w-wr$$ $$\...
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1answer
72 views

Differentiation calculation

$L(E)$ espace fonction continuous and linear $$\begin{array}{llll} \psi:& L(E)\times E&\longrightarrow& E\\ &(u,x)&\longrightarrow &u(x) \end{array}$$ proved the application ...
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3answers
67 views

Solving Second Order Linear Non-homogeneous Differential Equation

I am trying to solve the following: $$ y''+4y=\tan(t). $$ I have used the method of variation of parameters. Currently I am at a point in the equation where I have this: $$u_1= \int \frac{\tan t \...
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44 views

Decidability - Complexity

Can someone tell me where I can get some information about the following? We have linear differential equations with polynomial coefficients depending on x. $a_n(x)y^{(n)}+ \dots a_1(x)y^{(1)}+a_0(...
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1answer
139 views

Unique weak solution of Poisson's equation

Let $\Omega$ be an open set in $\mathbb{R}^n$ and now consider the weak formulation of Poisson's equation $$\int_{\Omega} \langle Du,Dv \rangle = \int_{\Omega}{fv}$$ for $v \in H_0^1$ and $u \in H_0^1$...
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56 views

two variable perturbation analysis of nonlinear set of differential equations.

I have following set of equations, $\frac{dy(t)}{dt}=k z(t) - 3 k y(t) - y(t)^2 + \epsilon_1 (M-z(t))^2$ $\epsilon_2 \frac{dz(t)}{dt}=Mz(t) - z(t) y(t) - 2 \epsilon_2 y(t) + 2 \epsilon_1 \epsilon_2 (...
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22 views

Recognizing and using hypergeometric function

Some expressions that interest me end up having something to do with hyper geometric function. I want to be able to derive such results myself. Where do I begin? For example, the equation $$ X''(x)\...
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54 views

Preparations to finals, validation needed

I have an exam in a few days from now and I'm very nervous. I tried to tackle this one with all I got, but I'm not sure if I'm correct. If anyone can direct me, and tell me if and where I'm doing ...
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0answers
37 views

Choose Scaling for t

My question is the last part of the d) part of the exercise 1.17 in Mark Holms' Introduction to Applied Mathematics. The exercise is given below, where I have emphasized the part of it that is my ...
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0answers
42 views

Sturm-Liouville eigenvalue problem of order 4

I want to solve the eigenvalue problem $W''''=\lambda W$ with the boundary conditions $W(0)=W'(0)=W(l)=W'(l)=0$. Has someone a hint how to solve that? Thank you...
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0answers
39 views

Phase line and Equilibrium Points

Consider the differential equation $dy/dt=y^8+3y^6-y^2-1$. Sketch the phase line and classify the equilibrium points. Since when $y=0$, the derivative is negative and when $y>1$ the derivative ...
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47 views

Non-Conservative System

I'm having a bit of trouble understanding the concept of a conservative system mathematically. A problem in a textbook (Arnold's Mathematical Methods for Classical Mechanics) is asking me to give an ...
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1answer
66 views

Make mathematical sense of the Dirac well Potential Equation

A classical problem in quantum mechanics involving the Dirac Delta function is given by $$ y''+(\delta(x)-\lambda^2)y=0 $$ Then, to find ''bound states'', you solve on the right and find the ...
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24 views

Regarding continuity and the value of the function at the point of discontinuity.

Suppose while solving a boundary value problem, we have a two piece solution $f_1(x)$ and $f_2(x)$ where $f_1(x)=f(x)$ for $x < x_0$ and $f_2(x) = f(x)$ for $x>x_0$. If there is a matching ...
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1answer
100 views

How do you read a partial differential equation?

In calculus we can read the "normal derivative", $\frac {df}{dx}$, as the rate of change of our function $f$ with respect to $x$. With partial derivatives of multivariate functions it is very much the ...
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1answer
121 views

Finite Difference for Hamilton-Jacobi-Bellman without boundary conditions

Let $t\in\mathbb{R}_+$ denote time, $x \in X$ is the state and $u \in U$ the control. The objective function is $F:X \times U \to\mathbb{R}$ and $f:X \times U \to\mathbb{R}$ is the law of motion for ...
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1answer
384 views

How can i find the basis solutions of homogeneous linear ODE?

Second order linear differential equation is given below. $y''+\frac{2}{x}y'+k^2y=0,$ where $k$ is constant and $x\neq 0$ I already know that the basis are $y_1=\frac{e^{-ikx}}{x}$ and $y_2=\frac{e^...
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20 views

Closed representation of Ladder operators in One Dimensional Second Order Homogeneous Differential Equations

(1) Has anyone published the closed representation of ladder operators for second order differential equations? More specifically the second order differential equation $$ -\partial_x^2\Psi_m(x) + V(...
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40 views

Initial value problem - How can we find the coefficients $c_j$?

We have the initial value problem $$u'(t)=Au(t) \ \ , \ \ 0 \leq t \leq T \\ u(0)=u^0 \\ u \in \mathbb{R}^m$$ A is a $m \times m$ matrix The eigenvalues of $A$ are $\lambda_j$ and the corresponding ...
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1answer
34 views

Differential spherical wave equation, why is the result the same for real and imaginary parts?

I wasn't very sure whether to ask this in the physics forum or here, but the question regards mathematics much more than it does physics. The following wave function is given (spherical wave): $\psi(\...
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35 views

Parabolicity of high order PDEs

I know that the traditional classification of PDEs into parabolic, elliptic, and hyperbolic is applicable for the second order equations. However, I often see remarks about parabolicity of higher ...
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40 views

All possible flat conformal metrics of dimension greater than 2

Combining List of formulas in Riemannian geometry and Conformal symmetry, is there a proof which states $$ x^\mu \to \frac{x^\mu-a^\mu x^2}{1 - 2a\cdot x + a^2 x^2} $$ represents all possible one-...
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62 views

Prove using Green's theorem that the boundary value problem has at most one solution

Prove using Green's theorem that the boundary value problem $$\frac{\partial}{\partial{x}}\left ( (1+x^2)\frac{\partial{u}}{\partial{x}}\right )+\frac{\partial}{\partial{y}}\left ( (1+x^2+y^2)\frac{\...