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

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A system of ODEs, what existence results are there?

Let $u(t) \in \mathbb{R}^n$. Are there existence results for the ODE $$C(t)u'(t) = A(t)u(t) + f(t)$$ where $A(t), C(t) \in L^\infty(0,T;\mathbb{R}^{n\times n})$, $f(t) \in L^2(0,T;\mathbb{R}^n).$ In ...
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61 views

Solving an eigenvalue problem on the open unit rectangle

Let $\Omega=(0,1)\times(0,1)$ and consider the boundary value problem $$\begin{cases}\Delta^2u=f\\ u(x,y)=\Delta u(x,y)=0,& x,y\in\partial\Omega \end{cases}$$ I want to solve this boundary value ...
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67 views

Ordinary Differential Equation and graphs theory?

Is there any application of Ordinary Differential Equation in graphs theory?
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38 views

eigen problem for direct scattering method

Consider the KdV equation $$u_{t}+6uu_{x}+u_{xxx}=0$$ with initial condition $$u(x,0)= \begin{cases} 1 &\text{if } x \in [-1,0] ,\\ 0 &\text {if } x \in ...
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57 views

Establishing bounds of differential equation using a maximum principle

I would like to establish that the solution of $$-\epsilon u''_\epsilon+b(x)u'_\epsilon=f(x)$$ satisfies $$||u^{(k)}_\epsilon||\leq C(1+\epsilon^{-k/2}),$$ where $b,f\in C^4(\bar\Omega)$, $b(x)\geq ...
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95 views

Exact Differential Equations of Order n?

A second order ode $Py'' + Qy' + Ry = 0$ is exact if $$(Ay' + By)' = Ay'' + (A' + B)y' + B'y = Py'' + Qy' + Ry = 0$$ How can one cast the analysis of this question in terms of exact differential ...
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68 views

Path traced by a water drop on an ellipsoid

If we have a smooth football in the shape of an ellipsoid, and that water runs down on its sides, can we trace the path of a water drop on it? For a sphere it seems easy because the force tangential ...
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116 views

solve $axy''-by'+cxy=0$ step by step

Solve $$axy''-by'+cxy=0$$ step by step I know the solution is $$y=k_1x^{u}J_{u}\left(\sqrt{\frac{c}{ a}}x\right)+k_2x^{u}Y_{u}\left(\sqrt{\frac{c}{ a}}x\right)$$ Where $k_1,k_2$ are arbitrary ...
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119 views

How can I solve this PDE using change of variables?

I am currently struggling with this PDE: $$ (xy-x)u_x-(y^2+2x^2)u_y=0 $$ with the boundary condition $$ u(0,0)=0. $$ I have tried expressing it as $$ \langle u_x,u_y\rangle \cdot \langle ...
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72 views

Solution to this Poisson equation

I am struggeling with the following PDE. Does somebody here know a solution on the whole $\mathbb{R}^2$ that goes to zero for r approaching infinity? $\Delta ...
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147 views

Where to start with this non-linear first order ode

I would like to study the following system non-linear ode system because I hope to gain some insight into the curvature of a related metric. \begin{align} (q'_1 + q'_2) &= ...
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100 views

why does a fractional differential equation have a unique solution?

Why must there be a unique solution to a linear constant-coefficient fractional differential equation of order $(n,q)$ with $\lceil\frac{n}{q}\rceil$ initial conditions? (All notation is as in Miller ...
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677 views

Hard Differential Equation. Please help.

first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
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136 views

Lambert Omega Function

I just solved a problem and I reached a point where I could no longer simplify the equation. Being as impatient as I usually am on a Friday, I plugged my final line of derivation into WolframAlpha and ...
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136 views

Calculate half life of esters

I'm trying to calculate the level of testosterone released from different testosterone esters. Here are some graphs of testosterone levels after single injections of 250mg of each ester. Testo U ...
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55 views

Search for a candidate function with specific properties?

Given the following expression: $$ \mathcal{F(p,c,r,s)} = \frac{c^2 p^2 \left(s f'(s)-2 f(s)\right)^2}{4 f(s) \left(c^2 f(s) \left(c^2 p^2 f(s)+s^2 \left(r^2-p^2\right)\right)+\left(-r^2-1\right) ...
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265 views

Fourier Transform of one variable in a two variable function.

I have a function in two variables, that satisfies the following PDE: \begin{equation} \frac{x-x_0}{x-x_1}\Psi_{xx}+\Psi_{yy}=0 \end{equation} Initially I did use Fourier series \begin{equation*} ...
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194 views

Diffusion in Spherical Coordinates with mixed BC

I have been working through the book "A Guide to First-Passage Processes" and wanted to branch out on my own doing a calculation similar to what occurs in chapter 6. My basic problem comes from the ...
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93 views

Does this type of bifurcation exist?

I've been checking out numerically an ODE model of a gene circuit. Just from simulations, it appears that once a parameter passes some critical value a stable fixed point splits into three other fixed ...
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53 views

Methodology to Solve a Riccati Equation

I am new to solving ODEs and need some help. I have the following SDE: $\frac{d \eta_t}{dt} = \sigma_\mu^2 - 2 \lambda \eta_t - \sigma^{-2} \eta_t^2$ $\sigma_\mu$, $\lambda$, $\sigma$ are ...
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Dynamics of solutions close to $x(0)$ of $\dot{x}=\sqrt{x}+f(t)$ for $f(t)$ small when $t \ll 1$

I was looking at the dynamics of the real solutions close to $x(0)=0$ for the non-autonomous ODE \begin{equation} \dot{x}= \sqrt{x} +f(t) \end{equation} where $f(t)>0$ is `small' for $t \ll 1$ ...
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207 views

This matrix is an attractor?

I'm trying to find for which values of $\gamma$ the matrix A is an attractor: $$ A=\begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ -1 & 0 & \gamma ...
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Finding the best real value for $C$.

Consider the recurrence $f_{n+1}=f_n + \ln(f_n)$ with $f_0=2$. Also consider differential equations of type $g(0)=2$ and $\dfrac{d g}{d x}=\ln(g(x)- C \cdot \ln(g(x)))$. Lets call the solution ...
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Looking for online matlab-based differential equations course/text.

I am looking for an online ODE course that would be matlab/project-oriented. A full online text/course in the spirit of this linear algebra text is preferred. I know about the following CODEE and ...
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80 views

How to solve two-level Schrödinger equation using Floquet theorem?

Consider a sinusoidal driving two-level system: $$ i \left( \begin{array}{c} \dot C_1(t) \\ \dot C_2(t) \\ \end{array} \right)=\left( \begin{array}{cc} -\frac{\omega _0}{2} & ...
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168 views

Method of undetermined coefficients for the input functions associated with the unit step

I am trying to solve a second order non-homogeneous differential equation where $x(t)$ has $u(t)$, the unit step as a part. i.e. $ x(t)= f(t)u(t) $ I know how to 'guess' the particular solution for $ ...
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162 views

Unusual jump condition for Green function

This question is related to a previous question I posted a while ago. Imagine that I'm computing the Green function of a linear operator $L$, such that: $$LG(x,s)=\delta(x-s).~~~~~~~~~~~(1)$$ Now, ...
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605 views

How to apply Duhamel's Integral

I found one good procedure for solving the simple system of two equations with reducing on Duhamel's Integral, but I have problem to apply the same procedure on system with four equations. Let's see ...
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I have an infinite solution to an ODE even though it has only a regular singular point

I have the ODE: $\displaystyle y''(x)+\frac{y'(x)}{x+1}+y(x)=0$ I know that this has a regular singular point at $x=-1$, as $(1+x)^{-1}$ has only a first order pole, and $1$ has no pole at all, and ...
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81 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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Differential Equation - $y'=5|y|^{4/5}, y(0)=0$

in the spirit of this question I ask about this one. $y'=5|y|^{4/5}, y(0)=0$ If $y> 0$ then $$y'=5|y|^{4/5}\iff y'=5^{-1}y^{4/5}\iff 5^{-1}y'y^{-4/5}=1\iff y^{1/5}=x+C\\ \iff ...
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Solving more complex diferential equations

I've come up with this implicit equation $ (y')^2(2x-2x^2+2y^2)+(y')^2=1 $ and I'd like to find the function $y(x)$ (so that it's definition doesn't contain it's derivative). Only thing I've been ...
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149 views

Show that this orbit has a zero Lyapunov exponent

I'm using J.Meiss -Differential dynamical systems, and have some trouble to understand a proof about Lyapunov exponents. We have a dynamical system $$ \dot{x} = f(x), $$ with the corresponding flow $ ...
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237 views

Hodograph transformation and implicit solution of a non-linear PDE

I am trying to understand how can one apply the Hodograph transformation to a non-linear PDE. I read that this transformation implies the representation of the solution in the implicit form . So, if I ...
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771 views

Existence of a unique solution with given initial value problems.

Directions: Find an interval centered about $x = 0$ for which the given initial-value problem has a unique solution. $$(x - 2)y'' + 3y = x$$ Initial values: $y(0) = 0,\,\,y'(0) = 1 $. My answer ...
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van der pol and liapunov

i have attempted this question and done as much as i possibly could, any help regarding this question would be very helpful and appreciated. a) show that the second-order differential equation for ...
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102 views

Continuity of the inverse Laplace Transform

If I know $Y(s)$, can I predict when $\mathscr{L}^{-1}[Y(s)]=y(t)$ will be continuous or continuously differentiable or even stronger conditions? For example; I'm solving an ODE with the Laplace ...
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280 views

Floquet's Theory, Hills Equation

Let us examine Hill's equation $\ddot x+Q(t)x=0$, where $Q$ is piecewise continuous and with a period $T$. Let $\mu_{1,2}$ be the multiplicators. Let $\lambda$ be the characteristic exponent. How can ...
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Property of first order differential equation

I need help with following exercise: Let $f$ be real function in $R$ of class $C^1$ and $f(r)=r$. Show that if $f'(r) \lt 1$ then no solution of the equation $x'=f(x/t)$ is tangent at $0$ to ...
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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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What are the connections between spectral expansion and differential operator?

For instance, for a nice function $f$ on the unit circle, we have its Fourier expansion, $$f(x)=\sum_n \hat{f}(n) e^{inx},$$ where the exponentials are eigenfunctions for differential operator ...
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86 views

Unusual 2nd order inhomogeneous equation..

For some research Im doing, I've derived an equation of the form below for $C(r)$ $$C'' + \frac{2}{r}C' = W + \frac{f}{C}$$ Or, if you prefer, $$CC'' + \frac{2}{r}CC' - W\cdot C = f$$ This has the ...
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Construction of a system differential equation from the projected 2D solution curves

! A 3 variable system.I've been give the behaviors on the 2 projected planes. How do I arrive at the final 3d curve? My question is a very simple one.I have a system of 3 variables:- X,Y,Z.I have ...
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50 views

Approximate Differential Equation?

Let $x \in \mathbb{R}$ be a variable and $c\in\mathbb{R}$ a parameter. Also, let $f(x,c)$ be a function dependent on $x$ and $c$. Furthermore, define a Differential Equation which is solved by the ...
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Generating Function for rational sequence

I'm trying to compute the generating function for the function defined for $1\le N < \beta$, $C_N = \frac{\beta}{N(\beta-N)}$. I think my math so far is correct, but I don't know how to solve the ...
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73 views

closed form solution of an ODE

I have a problem with finding the closed form solution of the following ODEs. Closed form here means that the solution can be presented as integrals/ power series. Here is the ODE : I only consider ...
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Invariant Subspaces and Differential Equations

Given I'm given a marginally stable system, $\dot{x}(t)=Ax(t)$, where$A=\begin{bmatrix} -1 & -10 & -10\cr 1 & 0 & 0\cr 0 & 1 & 0 \end{bmatrix}$, and $x(0)=x_o$.The eigenvalues ...
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ODE: continuous dependence on parameters

Is it true that the solutions of the problem: $$\begin{cases} \frac{\text{d}}{\text{d} s} [s^{2-2/N} u^\prime (s)] + \frac{\lambda}{c_N^2}\ u(s)=0 \\ u(\bar{s})=1\\ u^\prime ...
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112 views

phase portrait question

My DE course uses an online homework service to distribute and collect homework. One of the problems in this set is to furnish an autonomous DE consistent with the phase portrait below: I came up ...
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a nonlinear ode with an quotient

I'm curious about the nonlinear ODEs with an quotient including dependent variable : $$y''(x)+\frac{Ay(x)x}{1+y(x)}+Bx=0$$ Could you give me a clue on solving this equation explicitly?