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

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Causality in Dirac delta forced harmonic oscillator

If I take the simple forced harmonic oscillator equation, apply the Fourier transform to both sides, and assuming the forcing function is a Dirac delta function (at the origin) I get: $ F(s) = \frac ...
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85 views

Sturm-liouville problem, first eigenvalue

Any idea to solve the Sturm-Liouville Problem $$ -\cos^{2}(t)g''+n\sin(t)\cos(t)g'-(n+1)\cos^{2}(t)g=(\delta)g, $$ with $t\in[\epsilon,0]$, and boundary conditions $g(\epsilon)=g(0)=0$? We may ...
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117 views

A photon in expanding Universe (a snail on a tree)

I want to know how far a snail can reach in expanding universe. It has a constant speed c = 1 and tree is expanding at speed $v= H_0 D$, with Hubble constant $H_0 = 1$. Here D(T) is the distance of ...
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103 views

Differential equation $y'(t) = 1-y(t) e^{y(t)-1}$

I am interested in finding a clean explicit solution (if possible) to the differential equation $$ y'(t) = 1-y(t) e^{y(t)-1}, $$ where $0 \le t < 1$ and $0 \le y \le 1$. This can obviously be ...
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84 views

System of many non-linear (quadratic) first order O.D.E. (numerical strategy or simplification)

I have a large system (N>100) of equations $\frac{d\vec{P}}{dt}= A(t) + B(t) \vec{P} + \vec{P}^T C(t) \vec{P}$ where $\vec{P}$ is a vector of N functions of the variable t. What is the correct ...
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94 views

Stability of limit cycle

What can be said about the stability of the limit cycle for $r=1$ of the equation $$\dot{r}=(r^2-1)\cdot (2 r \cos(\phi) - 1), \dot{\phi}=1?$$ This is a problem posed in Arnol'd's book on ODEs. ...
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38 views

Pure differential equation whose solution is a siluroid?

I am trying to find a differential equation for the siluroid that DOES NOT contain explicitly $\theta$, $\sin\theta$, or $\cos\theta$, but only $\rho$, $\dot\rho$, $\ddot\rho$. The siluroid equation ...
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540 views

Solve a differential equation using Fourier series

Assume I have a second order differential equation $\ddot{x} = F(x,\dot{x})$ (or an equivalent equation of first order) and that I know there is a periodic solution to it (for simplicity's sake, ...
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630 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 ...
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30 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 ...
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49 views

Find $\alpha$ such that $y'=\sqrt{1+y^4}-|y|^\alpha$ has global solutions

How do I find $\alpha$ such that $y'=\sqrt{1+y^4}-|y|^\alpha$ has global solutions? For example, imposing $y'=0$ for $\alpha=4$ we get that for solutions with starting point in ...
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106 views

Existence Theorem for Geodesics

The text I am reading now defined geodesics to be those curves that satisfy the following differential equation: $\ddot{\gamma}^k(t)+\dot{\gamma}^i(t)\dot{\gamma}^j(t)\Gamma^k_{ij}(\gamma(t)) = 0$ ...
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64 views

Second order equations on manifolds

In my notes, the lecturer considers a smooth vector field $v: TM\to T(TM)$, with $M$ a smooth manifold. Let's write $$v(u,e)=((u,e), (a(u,e),b(u,e)).$$ It is said that $v$ is a second order equation ...
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264 views

Behaviour of $r'=r-r^3 , \theta'=(\sin\theta)^2+a$

What are the local and global behavior of solutions of $r'=r-r^3$ $\theta'=(\sin\theta)^2+a$ at the bifurcation value $a=-1$?
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377 views

Integrating angular velocity to obtain orientation

Suppose that $\gamma:[0,1]\to \operatorname{SO}(3)$ is a path in the space of orientation preserving rotations of $\mathbb R^3$. It is classical that we can find a corresponding $\omega:[0,1]\to ...
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95 views

Complex nonlinear differential equation

I have the following nonlinear differential equation: $$\ddot z(t)-\sin(z(t))=0$$ where $z(t)$ is a complex variable. The solution of the same equation with $z(t)$ real, is a function of Jacobi ...
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301 views

system of implicit nonlinear differential equations

Here I have a system of nonlinear differential equations: $ (M+2m)\ddot{x} + m(l_1 \ddot{\theta}_1\cos\theta_1 - l_1\dot{\theta}_1^2\sin\theta_1) + ...
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186 views

The linearization of a gradient vector field along a heteroclinic connection

A gradient vector field $X$ in $\mathbb{R}^n$ has two equilibria $x_1, x_2$. The vector field defines a cooperative dynamical system. The linearization about $x_1$ has one positive eigenvalues and ...
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183 views

IVP Perturbation With Small Non-Linear Term

EDIT: Sorry to bump this without having anything extra to add, but I still cannot reconcile my solution with what was asked (in (2)). Could someone with expertise in this subject take a look? I ...
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156 views

Prove that the first positive root of the solution to the Lane-Emden equation increases steadily with $n$.

Let $\lambda$ be the first positive value for which $y=0$ where $y(x)$ satisfy the following differential equation $$ y''+\frac{2}{x}y'+y^n=0,\qquad\text{where }n\in\mathbb{R},\ y(0)=1,\text{ and }\ ...
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92 views

What's this called? $\mathbb{C}[d/dx]$

The 'ring of differential operators wrt x' ? Thx.
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151 views

Check my solution - Modelling of a spring with Differential Equation

I am doing some work with differential equations. I have solved the following problem but am uncertain if I'm doing it correctly. Could someone look over it for me and check if I'm doing something ...
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344 views

Harmonic oscillator with stochastic forcing

It's well known that the solution of the differential equation: $$\ddot x(t)+\omega^2x(t)=\sin(\psi t)$$ has the form: $$x(t)=C_1 \sin(\omega t)+C_2 \cos(\omega t)-\frac{\sin(\psi ...
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165 views

Prove there are at least two periodic solutions

Could anyone comment on the following ODE problem? Thank you. Given a 2-d system in polar coordinates: $$\dot{r}=r+r^{5}-r^{3}(1+\sin^{2}\theta)$$ $$\dot{\theta}=1$$ Prove that there are at least ...
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404 views

Gompertz growth equation

:) Hi! I'm almost finished with a homework problem, but I cannot quite finish it. The problem is as follows: Given the Gompertz growth equation $$\frac{dN}{dt}=K(t)N(t),\ N(0)=N_0 \\ ...
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424 views

Lebesgue Line Integrals - Parametric Change of Variables

Consider the following Lebesgue integral in $\mathbb{R}^n$ $$ \int_C f(x) dx $$ Where $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is measurable and $C$ is a measurable subset of $\mathbb{R}^n$ that ...
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282 views

Approximating a system of differential equations as a Bézier curve

I am looking for a general transform to approximate the solution to an n-dimensional system of differential equations and initial conditions as a cubic or quadratic Bézier curve. Sorry if my ...
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101 views

Finding $\mathbf r(t)$ for the parameterized two-body equations of motion

I'm trying to understand the equations of two-body motion. Namely, given the position, velocity and mass of two orbiting bodies at time $t$, how can I explicitly find their position and velocity for ...
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343 views

Differential equation, eigenvalues and eigenfunctions

How does one find all the permissible values of $b$ for $-{d\over dx}(-e^{ax}y')-ae^{ax}y=be^{ax}y$ with boundary conditions $y(0)=y(1)=0$? I assume we have a discrete set of $\{b_n\}$ where they ...
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92 views

Special forms of ODEs

In my previous question, @Gerben suggested that it is more likely that WA recognizes an ODE in"Sturm-Liouville" form. Is there a reason for this particular form being preferred to the usual ...
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195 views

Inequality of ODE solutions

Says I have two (scalar) ODE: $u' = f(u,t)$ and $v' = g(v,t)$ where Both $f$ and $g$ are piecewise-continuous and locally Lipschitz, for existence & uniqueness of solutions $u(t)$ and ...
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85 views

Differential equation with some constraints

I'd like $\alpha,\beta,\gamma$ as functions of $t$, satisfying the following conditions: $$ \begin{align} \alpha+\beta+\gamma & = 0 \\ \sin^2\alpha + \sin^2\beta + \sin^2\gamma & = c^2 \\ ...
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362 views

Half-life versus relaxation time

Question: What is the exact relationship between half-life and relaxation time? I just wanted to nail down the difference/similarity between these two concepts. I did a web search, and even found a ...
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235 views

Series of nested double integrals

This is kind of a follow-up of my previous question. I'm investigating the following infinite series of nested two-dimensional integrals $$\sigma(t,t^\prime) = 1 - \int_{t^\prime}^t\mathrm dt_1 ...
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62 views

An estimate for the left point in a BVP

Let $\alpha \geq 1$. Suppose that for each $c\geq c_0>0$ there exists a point $\xi (c) \in ]0,1[$ s.t. the BVP: $$\begin{cases} [x^\alpha u^\prime (x)]^\prime +c\ u(x)=0 &\text{, in } ...
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21 views

Solution of Initial Value Problem

Solve the given initial value problem: $(x+y)^2dx+(2xy+x^2-9)dy=0$ and $y(1)=1$ I thought one way is to to put $x=X+h$ and $y=Y+k$ to make the equation homogeneous but it seems a bit complicated. ...
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46 views

How to analytically continue this function?

I was wondering if it would be possible to get an analytically continuation of the following function: $$ J(x) = \sum_{r=1}^\infty \ln(r)x^r $$ My attempt Consider the following: (1) $$ J'(x) = ...
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16 views

About parametric resonance

I am trying to solve the same equation with little difference: $\ddot{x} +w^2(t) x=0$. Where $w^2=w_0^2 [1+h\cos\gamma t]^2$, where $h\ll 1$ and $\gamma =2 w_0 + \varepsilon$. How can I prove ...
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91 views

External ballistics: Prove that the range is a concave function of the elevation

Consider a projectile moving in a plane. One of many different models for this problem is the following ordinary differential equation \begin{align} x''(t) &= -Ex'(t), \\ y''(t) &= -Ey'(t)- ...
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71 views

How do we know that an integral is unsolvable?

I am currently learning intro differential equations. I am confused how one knows that an ODE will not be solvable. It seems that for the most part, the equations becomes "unsolvable" about halfway ...
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Differential equations and Vector spaces

I was reading Cohn's book on Lie Groups.In introduction part he has given the motivation behind Lie Groups.It is like this If solution of the differential equation $\frac{dx_{i}}{dt}=u(t)$ is ...
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What are the possible parameter values in this ODE?

I'm trying to solve the following ODE: $$(1-x^2)\frac{d^2f}{dx^2}+\left(\frac{1}{x}-3x\right)\frac{df}{dx}+\left[\sigma-\frac{n^2}{1-x^2}-\frac{m^2}{x^2}\right]f=0$$ for $x \in [-1,1]$. We have, ...
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36 views

How to finish up this first order differential equation?

$$x^2y'-5xy = x^8\sin x, x>0$$ NOW WITH EDITS! I understand the steps involved in solving this problem. Step one is to get to the form: $$y'+p(x)y=q(x)$$ So I divide by $x^2$ to get $$y' - ...
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34 views

Differential Identity $\nabla\phi(t,x)\cdot f(x)=f(\phi(t,x))$

Let $\frac{d}{dt} x=f(x)$, where $f:\mathbb{R}^n\to \mathbb{R}^n$, be smooth, and we let the flow of the ODE be $\phi(t,x)$. Show that: $\nabla\phi(t,x)\cdot f(x)=f(\phi(t,x))$ This was in my ODE ...
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43 views

Mathematical Epidemiology

Okay so this is quite a long question involving a lot of background work I have done myself but there are quite a few holes I need filling in so I'll start from the beginning. I'm writing a report ...
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100 views

Given triangle ABC, how to move point B to a certain angle given that its new location lies within the direction of its old altitude.

I have a 2D coordinate system for 3 known points $A$, $B$, $C$. Given that I can only move point $B$, how can I compute for its new coordinate with a certain angle $\theta$ considering that its new ...
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Why can't we model periodic phenomena using a single autonomous differential equation?

I have the system below. It is used to model the interaction between predator and prey. $$x' = x-xy, y' = -y + xy$$ The solution curves are closed contours about the point $(1,1)$. I determined ...
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46 views

Using the variation of parameters method to find the particular integral of an ODE

I have the following ODE: $$ x(x+1)y^{''} + (2-x^{2})y^{'} - (2+x)y = 0 $$ Given in the question were two sollutions to the associated homogeneous equation, which were $y_{1} = \frac{1}{x}$ and $y_{2} ...
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What is meant by a “hyperbolic periodic orbit”?

At the end of the Wikipedia article on "hyperbolic sets" (https://en.wikipedia.org/wiki/Hyperbolic_set) there is a reference to a periodic orbit being "hyperbolic", i.e. a periodic orbit of a ...
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34 views

Equilibrium points of $\dot x(t)=-2\cdot x^3(t)$

The following differential equation is given: $$ \dot x(t)=-2\cdot x^3(t)\qquad x(t)\in\mathbb R $$ I am asked to find the equilibrium points of the system. By definition, the equilibrium points are: ...