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

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67 views

dropping a particle into a vector field, part 3

Okay, so I've been independently trying to study basic systems of differential equations as they relate to dropping a particle into a vector field. I have had two previous posts on the matter trying ...
3
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178 views

What Happens At An Equilibrium Point For An Autonomous First-Order Differential Equation.

Let $\frac{dx}{dt} =f(x)$ be an autonomous first-order differential equation with equilibrium point at $x_0$. a) Suppose $f'(x_0) = 0$. What can you say about the behaviour of the solution near ...
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206 views

A probable inspiring proof to Poincare lemma

Poincare lemma says if a smooth $p$-form $\omega$ is closed, then $\omega$ must be exact. Let's put it in another way, it says the solution of $d\omega=0$ is $\omega=d\eta$ for some $(p-1)$-form ...
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89 views

Singular solution to $(x+2y)y'=1$

I have a problem and I got most of the solution, but don't understand how to proceed. The problem is to solve: $$(x+2y)y'=1, \qquad y(0)=-1.$$ Here is my reasoning: Substitute $z = x+2y$. Then ...
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246 views

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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86 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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543 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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632 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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302 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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185 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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152 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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346 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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427 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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197 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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28 views

Proving that $\lim\limits_{t\to\infty} e^{At}x_0 + \int\limits_0^\infty e^{A(t-s)}b(s)ds=\vec{0}$

Consider $x'=Ax+b(t)$, a system of differential equations. Given that $A$ has negative real parts in all its eigenvalues, and that $\lim\limits_{t\to\infty} b(t) = \vec{0}$, I need to prove that ...
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43 views

Solution of $x'=Ax$ is not what it is supposed to be

Consider a system of ODEs $x'=\begin{bmatrix} 0 &1 \\ -1&0 \end{bmatrix}x$. Wolfram Alpha says that the solution is $x(t) = \begin{bmatrix} \cos t &\sin t \\ -\sin t& \cos t ...
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45 views

Finding when fixed point is hyperbolic

Consider the IVP for the $2$-dimensional dynamical system ($X=[0, \infty )^2$) $$\dot{x_1}=a-x_1-\frac{4x_1x_2}{1+x_1^2}$$ $$\dot{x_2}=bx_1 \bigg( 1- \frac{x_2}{1+x_1^2} \bigg)$$ for all $t \in I$, ...
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11 views

Finding a lower and upper bound for the first eigenvalue of a Sturm Liouville problem

Given the eigenvalue problem $y''+\lambda (1+x^2)y=0$, $y(0)=y(1)=0$, I need to find a lower and upper bound for the first eigenvalue $\lambda_0$ (that fits the eigenfunction that has no zeros in ...
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23 views

Methods for first order PDEs in higher dimensions

What are the possible known methods for solving first order PDEs in higher dimensions? Is there anything else besides the method of characteristic curves? In particular, I have four first order, ...
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50 views

A kind of Sturm-Picone theorem?

My question is very simple: Suppose $u,v:(a,b)\subset \mathbb{R} \to \mathbb{R}^+$ solve \begin{equation} (p(x)u'(x))'=-q(x)f(u(x)) \end{equation} \begin{equation} (p(x)v'(x))'=-r(x)g(v(x)) ...
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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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94 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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73 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 ...