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

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75 views

What allows us to break up $dy/dx$ in solving a separable differential equation?

Suppose you had a separable first order differential equation that can be written as $$ \frac{dy}{dx}=f(x,y)=g(x)h(y) $$ Rigourously, what allows us to rearrange this as $$ \frac{1}{h(y)}dy=g(x)dx? $$...
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43 views

Help with an ODE

I need some help, I have this ODE but can't solve it for $y(x)$, I try every method I know, but with no succes,please, somebody can help me? $(\varepsilon-x)y=y'(-x+y^2-2x^2)$ Thanks.
4
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78 views

A second order differential equation

How does one solve the following differential equation $y^{"}+xy^{'}+(1-x^2)y=y\sin x$? I don't know how to proceed?
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58 views

Solution for $\frac{a}{x} = \int_0^1 \frac{f(z)}{\left(f(x)+f(z)\right)^2} dz$

I am looking for the function $f(x)$ that solves $\frac{a}{x} = \int_0^1 \frac{f(z)}{\left(f(x)+f(z)\right)^2} dz$ such that $f(0)=0$. Even hints how to approach to this question would be very ...
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114 views

Non-Linear Ordinary Differential Equation in Fluid Dynamics

So while trying to model the physics of a rocket shot from the ground through the atmosphere, I came up with a second-order Non-Linear ODE of the form: $$ \ddot y + \dot y^2 e^y = f(t) $$ This is ...
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175 views

Operators such that $\langle Ax,x \rangle=-\langle x,Ax \rangle$

Let $X$ be a Banach space. We consider the differential equation: $$x'(t)=Ax(t), \ \ \ t\in\mathbb{R}$$ where $A$ is a bounded operator on $X$. If $X$ is a Hilbert space, and $x(t)$ is a solution of ...
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101 views

Solving second order nonlinear ODE

Having the following second order ordinary differential equation: $$ \ddot{x} = a \cos(x) $$ where, $a$ is a constant. What's an approach to solve this kind of equation?
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165 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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46 views

Wronskian different from zero and solutions of ODE.

Let $a_0, \ldots , a_{n-1}$ continuous functions in an interval $I$.Consider the equation $$x^{(n)} = a_{n-1}(t)x^{(n-1)}+\cdots+a_0(t)x. \tag 1$$ Let $\phi_1, \phi_2, \ldots,\phi_n$ $n$ are ...
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102 views

Scalar Autonomous Differential Equation?

What precisely is a scalar autonomous differential equation? I'm confused about what this precisely means, more so because we did not discuss this in any lectures nor is it, as far as I can tell, ...
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92 views

Determining whether the origin is an attracting fixed point for a scalar system

I have been asked to determine and prove the attraction properties of a continuous-time dynamical system, generated by the ODE \begin{equation} \frac{dx}{dt} =-x \end{equation} which gives the system ...
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81 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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183 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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65 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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122 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 \frac{d\phi}{dr}...
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48 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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49 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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148 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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221 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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394 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 k^...
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475 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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102 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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881 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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154 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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240 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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133 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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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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18 views

Show that a solution of the complex ODE, with analytic coefficients, is itself analytic

Given the complex second order linear ODE $W''(z) + p(z) W'(z) + q(z) W(z) = 0$ show that the solution $W$ is analytic under assumption that $p$ and $q$ are so too. I thought that if a solution ...
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60 views

An analytic method to prove a specific curve is closed

In my study of Hamiltonian dynamics I have come across a Hamiltonian dynamic system with a solution curve I know to be closed via computer and via intuition but I require a rigorous way to prove this, ...
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28 views

Dissipation term in wave equation

If we're given a string with mass density $\rho$ in units $\frac{M}{L^3}$ with constant cross-section $A$, tension $T$ in units $\frac{F}{L^2}$, and whose length is $L$; and then we assume that the ...
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21 views

WKB problem with 4 turning points?

I was recently given a problem that asked to find the solvability conditions for $$\epsilon^2y''=(W(x)-E)y;\quad y\rightarrow0\text{ as }|x|\rightarrow0$$ where $W$ was some piecewise linear, $``W"$-...
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35 views

Find the paths along which water will run down the sides of a smooth football.

A smooth football having the shape of a prolate spheroid 12 inches long and 6 inches thick is lying outdoors in a rainstorm. Find the paths along which water will run down its sides. (Differential ...
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47 views

Solve differential equation using 'variation of parameters'

Given the differential equation $L \ u = f$, with $$L \ u = a_2(t) \frac{d^2u}{dt^2}+a_1(t)\frac{du}{dt}+a_0(t)u$$ with $a_i(t)$ sufficiently smooth and $a_2(t) \neq 0$ for every $t$. Suppose that $...
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33 views

Derivative of solution to IVP

I have two IVPs as followings: $$ y'=f(y, t)+g(y, t) \tag{1} $$ $$ y'=f(y, t)\tag{2} $$ with the same initial condition $y(t_0)=y_0$. Both $y$ and $t$ are positive and bounded. Furthermore, $f(y, t)$ ...
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55 views

Problem in integral on the phase plane

I have this formula to calculate the period of a motion in the phase space (plan, in this case) along a phase curve. \begin{equation} T(E)=\int_{x_1}^{x_2}\frac{dx}{\sqrt{2(E-U(x))}} \end{equation} ...
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56 views

conversion from stratonovich SDE to Ito's form?

conversion of stratonovich SDE to Ito SDE (Where $\partial$ is differential in the stratonovich form and $d$ is in ito's form): $$\partial X_t=\sigma(X_t,t)\partial B_t+b(t,X_t)\partial t$$. ...
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57 views

How to solve this DDE: $f'(x-1)-f''(x)=g(x)$.

How can I solve $$f'(x-1)-f''(x)=g(x),$$ or similar delay differential equations (DDEs) where there's some function on the r.h.s.? Is there a general method. I tried the following: Suppose $f(x)=Ce^{...
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69 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 $\lim\...
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51 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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45 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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58 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 $$x'=y+x(1-a^2x^2-b^2y^2)$$ $$y'=-x+...
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53 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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58 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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42 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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60 views

Asymptotic analysis of non-linear ODE

I'm looking for references on the topic of asymptotic analysis of non linear ODE's of the sort $$ x'' + x'x = 0 $$ This specific case has an analytic solution (with some $\tanh(\cdots)$ involved) and ...
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39 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 $v_r/v_s$....
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26 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)) y(...
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52 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) = \...
3
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0answers
102 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 x+\gamma),\...