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

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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{ } ...
4
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63 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 ...
4
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121 views

Help me understand this differential equation solution

I found a differential equation in an old paper, where the solution is a bit hard to understand. Given this equation: $$\frac{1}{2} r^2 \left(\frac{d \phi}{dr}\right)^2 + c^2 \left(r ...
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47 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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48 views

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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143 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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209 views

Solving $ T' = 0 $ for distributions in $\mathbb{R}^n$

Denoting $ T \in \mathcal{D}'(\mathbb{R}^n) $ as distributions with $ T_f(\varphi) = \int_{\mathbb{R}^n} f\varphi\ dx $, I wish to prove the distribution solution of the equation $ T' = 0 $ ...
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385 views

Solving inhomogenous bessel equation

I have the following differential equation to be solved $\dfrac{d^2\psi}{dr^2}+\dfrac{d\psi}{rdr}+4\left(\omega^2-k_0^2-\dfrac{n^2}{r^2}\right)\psi=AJ_n^2(kr)+\dfrac{k}{r}J_n(kr)J_{n+1}(kr)-\omega ...
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464 views

Confused by a proof in Rudin *Functional Analysis*

I am reading Rudin's Functional Analysis and got quite confused by his proof of Thm 8.5, that is, the existence of fundamental solutions for differential operator $P(D)$, where $P$ is a polynomial. ...
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100 views

$u''+\frac{4}{x+1}u'+\frac{2}{\left(x+1\right)^{2}}u=0$ variational solution

This is a concept solution scheme derived from a particular example that I have not been able to generalise sufficiently. The objective is to find a particular solution to a certain second-order ...
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2k 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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855 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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152 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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238 views

A solution of $-y'' + q(x)y= \lambda y$

Could you help me with the following problem (from Poschel and Trubowitz)? I am looking for a solution of the differential equation $-y'' + q(x)y= \lambda y$, for $0 \leq x \leq 1$ with ...
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121 views

Steady-state of `degenerate' delayed differential equation

Consider the simple delayed differential equation: $$X'(t) = -a X(t) + a X(t - d)$$ where $d$ and $a$ are positive constants. I'm interested in the possible steady-state (stationary) solutions of this ...
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176 views

How to analysis the stability of these ODE?

Study whether the null solution of the system: $$\begin{cases} \frac{dx_1}{dt}=x_2(t)\\ \frac{dx_2}{dt}=-w(t)^2 x_1(t)\\ \end{cases} $$ is Lyapunov stable, where $$ w(t)= \begin{cases} 0.4 ...
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98 views

Efficiently solving a large, sparse linear system $M(s)ab(s)=c(s)$ (determined by smooth functions) over some range of $s$

I'm looking at a differential equation on the edges of a graph (the application is neuroscience), and the Laplace transform of the solution on most of the edges has a general solution more-or-less of ...
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48 views

Resolving ODE-1 $(x^2 + y^2 +x)\,dx + xy\,dy=0$ am I wrong or my teacher is?

This is how I've resolve this ODE-1 : $$(x^2 +y^2 +x) \, dx + xy \, dy=0$$ Check if the eq is exact: $${\partial M \over \partial y}={\partial \over \partial y}(x^2 +y^2 x)=2y$$ $${\partial N \over ...
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42 views

Problem: differential equation

Hi I try solve the following problem of differential equation $$ x''+tx'+\frac{1}{1+t+t^2}x=0\tag 1$$ when $$x(1)=0\ \ \ ;\ \ \ x'(1)=1 $$ is the solution analytic in $t_0=1$ and his convergence ...
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28 views

proof that a system has at least one limit cycle

The problem is: Assuming that the parameters $a, b$ are real numbers and that $ab \neq0$, by transforming the system using polar coordinates, prove that the system $$x'=y+x(1-a^2x^2-b^2y^2)$$ ...
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37 views

Problem with the proof of the undetermined coefficients method

I'm trying to proof the following property which is the base of undetermined coefficients method. Say I have a differential equation of the form: $$x^{(n)}+a_{n-1}x^{(n-1)}+\cdots+a_0 = f(t)$$ When ...
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55 views

Trying to understand the concepts in ODEs text

I'm looking through a textbook on ODEs and the first chapter is basically just definition after definition after definition without very many exercises or examples for me to get my head around the ...
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26 views

How does one in general analyze the convergence of the following series?

The following question is inspired in the following videos: https://www.youtube.com/playlist?list=PL43B1963F261E6E47 Say one has a general second order linear differential equation $y''+Qy=0$ for ...
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27 views

Numerical method for SDEs

I'm using a 4th order Adams predictor-corrector method to numerically solve a regular differential equation. Now I would be interested to be able to include a noisy term to the equation -as in the ...
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32 views

Differential Equations - Crossing a River

I am an independently studying student, and I am trying to solve the question in the attached link. Essentially, I need help solving Equation (3) in the below attached picture for the ratio ...
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16 views

Arbitrary factors for the (modified) Mathieu equation

I am currently confronted with a physical equation that, after a fair amount of reworking, can be recast in the form of the modified Mathieu equation : \begin{equation} y(x)'' - (a - 2q \cosh(2x)) ...
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51 views

How to integrate this ODE?

How to solve the following ODE: $$ y'(x)=\sqrt{\dfrac{1-x+y}{2x+y}}. $$ I tried to transform it to a homogeneous differential equation, I found $$ s'(x)=\sqrt{\dfrac{s-1}{s+2}}-s$$ where $$ s(x) = ...
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94 views

Solve Burger's equation after shock forms

Solve the Burger's equation: $$u_t+(\frac{u^2}{2})_x=0,\quad 0<x<2,\quad 0<t<\infty,$$ with periodic boundary conditions and the initial condition $$u(x,0)=\alpha+\beta\sin(\pi ...
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48 views

Solving an integral within an integral

I have been trying for a while now to solve the following integral: $$ \int_0^t e^{-ct} \left(\int_0^t(1-M_r t)^a e^{ct} dt\right) (1-M_r t)^{-(a+1)} dt $$ I know the answer is: $$ \frac {1}{M_r^2} ...
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67 views

Two Matlab ODE solvers, two different results

I am solving a system of ODEs using Matlab. One particular set of parameters caused the solver to fail, so I worked my way through the different solvers Matlab provides. I was surprised to find that ...
3
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57 views

Center Manifold Exercise (small solution for small changes of the parameter)

Hi I'm stuck with this problem at first I didn't know how to begin so I copy an argument from [Carr, Application of Centre Manifold Theory]. But I don't know how can I find the coefficient from a, b ...
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102 views

Differential Equations: Jordan Form of a Matrix

I am using Lawrence Perko's book Differential Equations and Dynamical Systems, for my Differential Equations course. At the moment we are going over Jordan Forms of a linear system $x^{'}(t) = Ax$, ...
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72 views

Proofs from Ch. 1 of Arnold's ODEs

I've started reading Vladimir Arnold's Ordinary Differential Equations on my own. I like it so far, the only problem is that all of the exercises (as yet) are of the type "prove $X$" and without an ...
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21 views

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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50 views

Squares and constants in the dynamical system

I have $$ \begin{eqnarray} x'&=& x^2 - y^2 -1 \\ y'&=& 2y \end{eqnarray} $$ How can I solve such a system? I have tried the substitution $X= x^2 - 1$ but I still get constants in ...
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41 views

Coupled partial differential equation, with boundaries specification

Please, help me to find a books or samples to learn how to solve such coupled equations $$\begin{eqnarray} \frac{\partial T_1(x,t)}{\partial t}&=& \alpha_1 \frac{\partial^2 T_1(x,t)}{ ...
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57 views

Green's Functions: Solvable non homogeneous Sturm-Liouville with non homogeneous boundary conditions

I was just presented with this problem in my PDE Methods course which involves a non homogeneous Sturm-Liouville problem, which states as follows: Find the conditions under which the following SL ...
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47 views

Stability of origin of dynamical system

Usually you can note some nice structure in the problem which enables construction of a nice Lyapunov function. But this one is just a monster. Maybe there is a trick I've missed? Investigate the ...
3
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73 views

How to solve this first order diff. equation?

I'm looking for an analytic solution to a first order non-linear differential equation that I'm unable to solve : $$\frac{2}{1 + \Theta^2} \; \frac{d\Theta}{d r} = 1 - \frac{1}{r (\Theta - r)}$$ ...
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47 views

Reducing a higher order nonlinear ODE to a system of first order ODEs

The ODE that I am trying to reduce is: $$ y''' + 4\,y'' + y' + 6\,y - 2y^{2} = 0 $$ I start by letting $$ y = y_1 $$ $$ y' = y_2 $$ $$ y'' = y_3 $$ $$ y''' = y_4 = 2y_1^2 - 4y_3 - y_2 - 6y_1 $$ ...
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52 views

Using a Fourier Series to Solve Differential Equation

The problem states to use the fourier series of the function f(t) defined as follows: $f(t)= t+1 , -1<t<0 $ $f(t)=1-t , 0<t<1$ to solve the differential equation: x''+4x=f(t), x(0)=1, ...
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23 views

Which functions solve autonomous ODEs?

Fix an open set $U \subseteq \mathbb{R}$. What can be said about the set $a_n(U)$ of functions $f \in C^n(U, \mathbb{R})$ for which there exists a (sufficiently nice) nonzero function $g : \mathbb{R} ...
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81 views

Differential Equations which involve Infinite Series

The problem statement is as follows: Find the general solution for the following equation for $x(t)$. $$x''+ 9x = 2 + \sum_{n=1}^\infty \cos(nt)/n^3$$ I can't find anything about this in my ...
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71 views

Find extremum of functional

I want to find the extremum of $$J(y)= \int_1^2 \frac{\sqrt{1+y'^2}}{x}dx, \ y(1)=0, \ \ y(2)=1$$ I thought to use the following theorem: If $y$ is a local extremum for the functional $J(y)= ...
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36 views

Uniqueness of Rectifying Coordinates: Question for Arnold's ODE Book

In section 7 of his book Ordinary Differential Equations, VI Arnold explains the `rectification theorem', that, given an ordinary differential equation $$\dot{\mathbb{x}} = \mathbb{v(x)}$$ where ...
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75 views

Solve $x(x-1)y''+6x^2y'+3y=0$ using Frobenius's Method

Solve $x(x-1)y''+6x^2y'+3y=0$ using Frobenius's Method I can't solve this ODE. How can I get first two term? and indicial equation is also very confusing. I can solve two term recurrence ...
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79 views

What type of equation is this?

Is this equation an ODE or PDE $$ \frac{d^3u}{dx^3}−αxu=0, x∈R $$ The only thing given is $\int_R u(x) =\pi $ and $α>0$ is some constant. I have to find the solution using fourier ...
3
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82 views

If $y'=\frac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2) $

If $y'=\dfrac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2) = ?$ By integrating I am getting $$y = \ln (x+1)+C$$ I am stuck somewhat as it looks tricky from here. Any help ? Thanks!
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309 views

Solving a differential equation with a square root

I am trying to solve the differential equation $ A(x)\frac{d^{2}f(x)}{dx^{2}}+B(x)\frac{df(x)}{dx}=\frac{1}{3}\frac{1}{\sqrt{f(x)}}, $ where $ A(x)=\frac{x}{x+1} $ and $ B(x)=\frac{2x+1}{(x+1)^{2}} ...
3
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0answers
157 views

Wave equation for a string nonuniform (PDE)

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman , but I have been impossible these paragraphs. The displacement $u$ of a nonuniform string ...