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

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47 views

Show that this initial-value problem has a unique solution

I am trying to show that the following initial-value problem $$\frac{dx}{dt} = - x + tx^{1/2}; \quad x(2) = 2$$ has a unique solution on $I = [2,3]$. By letting $f(t,x) = - x + tx^{1/2}$ and $(t_0 ...
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64 views

How to integrate/differentiate parameters in differential equations

I have an ODE, which I would be fine about solving, were it not for the parameter: $$(\omega^2+x^2)\frac{dy}{dx}=y$$ I'm given that $\omega>0$ is a parameter. Separating the variables gives: ...
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46 views

Would the transformation of a differential equation obey the same algebra?

I've found that the algebra of this differential equation $$\frac{d^2y}{dz^2}-(3z^2+\gamma)\frac{dy}{dz}+(cz+\alpha)y=0$$ is in $sl(2)$ because it is possible to use the generators of the $sl(2)$ ...
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84 views

Measurability of points regular

I'm reviewing the proof of the theorem of oseledet the book Mañe: Let $M$ a compact metric space and $f:M \rightarrow M$ a homeomorphism, $\pi: F \rightarrow M$ a finite-dimensional continuos vector ...
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61 views

Integral curves in neighborhood of a minimum point of the potential

Let $X: \mathbb R^3 \rightarrow \mathbb R^3$ be a smooth vector field, and $f: \mathbb R^3 \rightarrow \mathbb R$ be its potential (i.e. $X= grad\ f$). When $f$ has a minimum point $p_{min}$ and all ...
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72 views

Quadratic system of ODEs

I have a quadratic ODE system that looks like this: $\dot{x}=Ax+diag(x)Nx$ where $x \in R^n$ and $A,N \in R^{n \times n}$ and $diag(x) \in R^{n \times n}$ is a diagonal matrix in which $x$ is its ...
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88 views

Does the implicit function theorem imply Peano existence theorem

In The implicit function theorem written by Krantz & Parks, it's said that the implicit function theorem implies the following existence theorem of ODE: Theorem 4.1.1 If $F(t,x)$, ...
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45 views

Qualitative dependence of solution to second-order matrix differential equation on eigenvalues

Suppose we have a matrix differential equation in $\vec{x}(t)=\left(\begin{smallmatrix}x_{1}(t) \\ \vdots \\ x_{n}(t)\end{smallmatrix}\right)$, such that: ...
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110 views

ODE with delta function

Consider the following ODE $$y''+a\delta (x)y+\lambda y=0$$ subject to the initial conditions $$y(\pm\pi )=0$$ (1) Show that there is a set of eigenvalues $$\tan (\pi \sqrt{\lambda ...
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45 views

Merton's Problem Stochastic Differential Equation

Solve the following numerical case of Merton's optimal portfolio selection problem: find an optimal policy function $(s, y) \mapsto u(s, y)$ such that for the Ito diusion determined by $dX_t =X_t(u(t, ...
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71 views

solve nonlinear second order ODE

I obtained Nonlinear second order differential equation as $y\cdot y''+y'^2-m\cdot y^{-a}y'^2+k=0$, Where $y'= \dfrac{dy}{dx}$, $y''=\dfrac{d^2y}{dx^2}$. I could not obtain the solution so please ...
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78 views

Rigorous separation of variables.

Let $I \subseteq \mathbb{R}$ denote an interval and $f$ and $g$ denote functions $$f : I \rightarrow \mathbb{R}, \;\;g : \mathbb{R} \rightarrow \mathbb{R}.$$ Now suppose we're interested in finding ...
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129 views

Lyapunov Function and $\omega$-limit sets

I will ask you about a particular equation but what I would really enjoy is an (if possible, comprehensive) answer to the following question : How can we, using a Lyapunov function, study the ...
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56 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 ...
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123 views

Prove that all the solutions of (2): $\frac{dy}{dt}=A(t)y+f(t)$ are bounded in $ \left[t_0,+\infty \right )$

I have a problem: Assume that system (1): $$\dfrac{dx}{dt}=A(t)x$$ is stable, where $A(t) \in C\left [t_0,+\infty \right )$, when $t \to \infty$ and $$\begin{cases} & \mathrm{ } ...
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55 views

Can Fredholm integral equation of the first type be represented as a differential equation?

Can Fredholm integral equation of the first type be represented as a differential equation? In other words, given a Fredholm integral equation of the second type does there exist a differential ...
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181 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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72 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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187 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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41 views

Perturbation Theory on Finite Domains

In this video (from 27.00 - 50.00, which you don't need to watch!) a guy shows how you can solve the general second order ode $y'' + P(x)y = 0$ using perturbation theory. However he points out that ...
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72 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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111 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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100 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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75 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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738 views

Questions about the Picard–Lindelöf theorem for an ODE

In the sketch proof of Picard–Lindelöf existence theorem for an ODE: It can then be shown, by using the Banach fixed point theorem, that the sequence of "Picard iterates" $φ_k$ is convergent and ...
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412 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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395 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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28 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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41 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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95 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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60 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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198 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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327 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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68 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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266 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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292 views

Chebyshev Diff EQ

Find a power series solution about $x_0=0$ for the Chebyshev differential equation $$(1-x^2)y''-xy'+n^2 y=0,$$ as a function of of the integer $n$. Show that the solutions form a terminating ...
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123 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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140 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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87 views

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

The 'ring of differential operators wrt x' ? Thx.
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252 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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1k views

Is it possible to have Wronskian=0 with independent solutions to a linear differential equation?

In Wikipedia it says that if the Wronskian of two function is 0 everywhere it does not imply they are linearly dependent. However, in books treating differential equations it seems that, if the two ...
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144 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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370 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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758 views

Solving geodesic problems with Euler-Lagrange equation

This is the question: Problem B.1 Two cities - Tel-Aviv, Israel and SanDiego, CA - have the same latitude 32 ◦ N, but, different longitudes: Tel-Aviv is 34 ◦ E and San-Diego is 117 ◦ W. What is the ...
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138 views

Satisfying a Differential Equation and complex Laguerre

I have the following problem Show that $$L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$$ satisfies $$x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$$ where the contour is ...
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360 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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230 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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100 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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313 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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79 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 ...