# Questions tagged [differential-equations]

Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.

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### Complex delay differential equation with time-dependent lag

I am trying to find a solution $g$ to the following delay differential equations (DDEs): $$\beta(t)g^\prime(y)=g(y)-g(y-t)-t \quad (1)$$ $$\beta(t)g^\prime(y)=g(y+t)-g(y)+t \quad (2)$$ with $g(y)=0$,...
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### Convergence of the solution to some stochastic equation''

For every $\epsilon>0$, consider \begin{equation*} X^{\epsilon}_t = Z + W_t - 1 + \mathbb E\left[\exp\left(-\frac{1}{\epsilon} \int_0^t \max\left(-X_s^{\epsilon},0\right)ds\right)\right],\quad \...
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### Initial conditions in the Klein-Gordon equation

I am interested in what are the largest families of initial conditions of the problem (in $\mathbb{R}^{4}$) \begin{equation}\label{kg} \left\lbrace \begin{array}{ll} (\square+m^2)F(x)=0\\ ...
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### A continuity argument for a dispersive $gKdV$ estimate

I'm learning about the gKdV equation, following Schlag & Muscalu vol II. We're looking at $$\begin{cases} u_t + u_{xxx} + F(u)_x = 0 \\ u_0 = g\end{cases}$$ where $F(u) = u^5$ (for example). The ...
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### Find strictly subharmonic function that vanishes at infinity

I am not sure about the term "strictly" subharmonic. What I want is a function $\psi\in C^{\infty}(\mathbb{R}^3)$ with $\Delta\psi>0$ and $\lim\limits_{|x|\rightarrow\infty}\psi(x)=0$. I ...
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### Solving the Airy equation by Borel summation

The Airy equation is the canonical example of the Stokes phenomenon but, as of yet, I've not seen it being solved by Borel summation (which is the main way to explicitly construct examples of Stokes ...
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### Solving a differential equation related to the hypergeometric differential equation

I need to solve the following equation: $x*(1 - s*x) y''[x] + y'[x] + r*y[x] =0,$ where $s$ and $r$ are two parameters. It would seem that is similar to the hypergeometric differential equation, but ...
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### Is there a general solution for the differential equation $f''(x) = f(f(x))$?

I'm currently an undergraduate studying differential equations and I've been fixated on the differential equation $f''(x) = f(f(x))$ for the past 2 days. I can't seem to crack it but it feels like it ...
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### Modelling stochastic solutions for the general stochastic epidemic

I have found the deterministic solutions for the following system of differential equations: $$s'(t) = -\beta s(t) i(t) \\ i'(t) = \beta s(t) i(t) -\gamma i(t)\\ r'(t) = \gamma i(t)$$ So the ...
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### Existence theory for first order scalar discontinuous ODE

Consider the scalar i.v.p. in ${\mathbb R}$ $$x'=f(t,x), \; t\in[0,T], \; x(0)=x_0,$$ where $T\in {\mathbb R}$, $T>0$, $x_0\in {\mathbb R}$, and $f:[0,T] \times {\mathbb R}\mapsto {\mathbb R}$...
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### Is it possible that $\int |f^2(z)|^{t+1}(P\bar{P})\phi(z) dz=0$ for all compactly supported $\phi$?

When proving that the log-canonical threshold is minus the largest root of the Bernstein-Sato polynomial one considers the integral $$\int |f^2(z)|^{t+1}(P\bar{P}(\phi(z))) dz,$$ where $P$ is a linear ...
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### Beltrami equation with harmonic coefficient

I need to find solutions to the Beltrami equation $$\frac{\partial w}{\partial\overline{{z}}}=e^{i\varphi(z)}\frac{\partial w}{\partial z}$$ for $w=w(z,\overline{z})$ and $\varphi(z)$ some given, ...
Suppose I have the following nonlinear coupled dynamic system \begin{align*} &\dot{x}_1 = f_1(x_1,x_2)\\ &\dot{x}_2 = f_2(x_2) + u \end{align*}where $x_1\in \mathbb{R}^{n_1}$, $x_2\in \mathbb{... 1answer 58 views ### Are two closed curves similar to each other if they have the same tangential vectors at the same position$\xi \in [0, 2\pi)$? Given two closed curves$X,Y:[0,2\pi) \rightarrow \mathbb{R}^2$with$X(0)=X(2\pi)$and$Y(0)=Y(2\pi)$, if their tangential vectors are the same, which means$\mathcal{T}(X)|_\xi=\nabla_\xi X(\xi)/|\...
This popped up in my research: we have the following mix-effect inequality that $\forall T \geq 1$ \begin{equation}\tag{*} Y(T) - \frac{1}{100T^2}\int_1^T[\alpha^2 + e^{-(T - t)}]Y(t)dt \lesssim \...