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

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Solving 1D telegrapher's equation by reduction to two-dimensional wave equation

The solution $w : \mathbb R \times \mathbb R_{+} \to \mathbb R$ of the Cauchy problem for the telegrapher's equation $$ w_{tt} - c^2 w_{xx} + c^2 \lambda^2 w = 0 $$ with $$ w(x,0) = 0, \qquad ...
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51 views

Finding the exact solution of a differential equation

Let $y=f(x)$. Is it possible to find an exact solution of the following differential equation?: \begin{equation} \ddot y+2\dot y-5xy=e^{-2x}\nonumber \end{equation} Many thanks in advance, -- Cesar ...
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198 views

Estimating a dynamical system's behavior without using Liapunov theorem

Assume that we have the following dynamical system $$x'=(\epsilon x+2y)(1+z)$$ $$y'=(-x+\epsilon y)(1+z)$$ $$z'=-z^3$$ Then how can I show that any solution that started from the region $z>-1$ ...
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34 views

Help proving(well, disproving) the convergence of $\sin^3(x)$

So I'm stuck on a question where it's asking for the power series and radius of convergence of $\sin^3(x)$ I've done the power series ok, but my problem is that when I apply the ratio test it's ...
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26 views

Green's function the way George Green defined it

This is a curious question about the way George Green could have defined his Green's function. All the definitions I see have only Dirac-delta $\delta(x-x')$ function as their source on the RHS. But ...
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47 views

Door mechanism differential equation

I have been wondering about a door mechanism I have seen. It has a wire attached to the upper corner of the door and from there to the corresponding corner in the door frame, where a weight hangs from ...
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56 views

How do I solve this differential equation?

$y^{(7)}+4y^{(6)}+8y^{(5)}+9y^{(4)}+8y^{(3)}+8y^{(2)}+8y^{(1)}+4y=e^{-x} (5sinx-cosx) $ The characteristic equation $ \lambda ^7 +4\lambda ^6+8\lambda ^5+ 9\lambda ^4 +8\lambda ^3+8\lambda ...
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19 views

Lipschitz function in ODE

Let $f:\mathbb{R} \times \mathbb{R^n} \rightarrow \mathbb{R^n}$ lipschitz function. Show given that $(t_0,x_0)\in \mathbb{R} \times \mathbb{R^n} $ there exists a unique solution: $$x' = f(t,x)$$ ...
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51 views

Fundamental Solution of a Nonlinear ODE (using Riccati Transformation & Wronskian)

I am given the differential equation: \begin{equation*} y^{\prime}(t) = y(t)^{2} + 2\sin(t)\cos(t) - \sin^{4}(t) \end{equation*} and one solution $y_{1}(t) = \sin^{2}(t)$. I wish to find a second ...
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67 views

Cauchy-Euler Equation of order $n$

What I wish to prove is that for a Cauchy-Euler equation of order $n$, the substitution $x=e^{t}$ transforms it into a linear differential equation with constant coefficients. To put it as a theorem: ...
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67 views

Differential Equation has a unique solution periodic

Let $A(t)$ continuous and periodic of period $S$ in $\mathbb{R}$. Suppose $x' = Ax$ has $\varphi \equiv 0$ as the only periodic solution of period $S$. Show that there exists $\delta> 0$ such that ...
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23 views

Invertible e converges series.

If T is a linear transformation on $R^n$ with $||T - I||<1$, prove that $T$ is invertible and that the series $\sum_{k=0}^\infty(I-T)^k$ converges absolutely to $T^{-1}.$ (Use the geometric series) ...
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58 views

WKB and asymptotic behavior of second order differential equation

I want to study the large $x$ solution to a Riccati equation. After listening to the lectures on Mathematical Physics by Carl Bender, I have fallen in love with asymptotic analysis. But, by no means ...
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56 views

Are any solutions lost when solving non-exact differential equations?

I have just started studying differential equations, one of the problems I found while I was practicing is "Consider the equation $$ (y^2 + 2xy)dx - x^2 dy=0 $$ (a) Show that this equation is not ...
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75 views

Zeros/poles at Laplace and at Fourier Transform

I recently started "relearning" the Laplace transform, and I noticed something. It seems to me that the intuitive idea of poles and zeros is different between these two transforms! For example, in ...
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58 views

Solve ODE by Fourier transform, and versus by Laplace transform?

Regarding solving ODE by Fourier transform, I read a nice reply by O.L.. After applying Fourier transform to an ODE to obtain an algebraic equation, the reply showed that some terms involving the ...
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33 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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50 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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55 views

Dynamical Systems: Attractivity and Stability

I have been asked to determine and prove the attraction properties of a continuous-time dynamical system, generated by the ODE $$dx/dt=-x$$ which gives the system $$S_tx=x_0e^{-t} $$ For the compact ...
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43 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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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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57 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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69 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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71 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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38 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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171 views

The method of Frobenius and the hypergeometric differential equation

Wikipedia has a nice article on how to use the method of Frobenius to find the the series solutions to $$x(1-x)y'' + \Big( \gamma -(1+ \alpha + \beta)x \Big)y' - \alpha \beta y = 0$$ near the ...
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80 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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40 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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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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95 views

Analytic solutions to a nonlinear second order PDE

I am trying to solve the following 2nd order PDE analytically, but haven't succeeded so far. I have tried to separate the variables, but it doesn't work here. I would be very grateful for any ...
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67 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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97 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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91 views

Prove that a substitution reduces the ODE to a separable equation

Show that the substitution $y=at+bx+c$ changes $x'=f(at+bx+c)$ to a separable differential equation. I did the proof. I want to check if it is correct, but appart from checking that, I have a doubt, ...
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51 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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65 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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137 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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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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69 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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108 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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99 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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Analog of Picard's theorem for Fractional Differential equations.

I need an analog of Picard's theorem of existence and uniqueness of solutions. The theorem is to be applied to linear fractional order differential equations with constants coefficients. I don't want ...
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556 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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349 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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99 views

Bifurcation in 3 dimensions (simple)

I am Doing a project i have a toy system that describes a bifurcation in 3 dimensions i am posting this in part because i can no longer understand what i have written down ( its been awhile) i have ...
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260 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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27 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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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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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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142 views

The simplest delay differential equation

I am trying to understand a bit about solutions of delay differential equations, so I tried analyzing one of the most simple ones: $$u'(t)=-\beta u(t-1), \text{and for } t\in [-1,0), u(t)=\phi(t), ...
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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 ...