# Questions tagged [differential-equations]

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

1,082
questions

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### How to solve or analyse the smallest eigenvalue of 2 coupled 1st-order linear ODEs?

I am trying to solve the eigensystem of a 1st-order linear ODE system in the region $(-\infty,\infty)$ and with Dirichlet boundary condition at the infinities
\begin{align}
-\mathrm{i} u'(x) +f^*(x) ...

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223 views

### Asymptotics of solutions to the ODE $f''(t)-e^{-2t} f(t)=0$

Consider the ode
$$
f''(t)-e^{-2t} f(t)=0.
$$
What is the general behaviour of $|f|$ for large $t$s?
Is it true that there exists an $A\in \mathbb{R}$ and there exists a positive $B$ such that $|f(...

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**1**answer

138 views

### Solution of nonlinear second-order ODE $y''+\frac{(y'+2ax)^2+4b^2}{2y}+\frac{10}{3}a=0$

Is there any way of solving the following second-order ODE
$$y''+\frac{(y'+2ax)^2+4b^2}{2y}+\frac{10}{3}a=0,$$
where $a$ and $b$ are some constant?
If we know that one solution exists, how would it ...

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### Second order non-instantaneous impulsive evolution equations

The first order linear non-instantaneous impulsive evolution equations is given as;
$u'(t)=Au(t)~~ t\in[s_i,t_{i+1}],\,i\in\mathbb{N_0}:=\{0,1,2,...\}$
$u(t_i^+)=(E+B_i)u(t_i^-),\,\,i\in\mathbb{N}:=...

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44 views

### Annihilator of the of the generating function not holonomic

The following is a generating function in $x,h$ with infinite parameters
$q_1,q_2\ldots,$ and $w_1, w_2,\ldots$.
$$\Psi(x, h)= \sum_{d=0}^{\infty} s_{(d)} (q_1, q_2, \ldots) \exp \bigg( \sum_{r=1}^{\...

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**1**answer

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### Holder inequality with respect to convex function

Given $a, T>0$, by Holder's inequality we have
$$
\int_T^{T+a}f(s)ds\leq \left(\int_0^{T+a}|f(s)|^2ds\right)^{1/2}\cdot\sqrt{a}.
$$
Do we have similar result if we replace $|x|^2$ by some convex ...

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**0**answers

57 views

### Keeping track of limit cycles via certain second order differential operator

Inspired by the two posts which are linked bellow we ask the following question:
Question: For a vector field $X$ on the plane we define the differential operator $D$ on $C^{\infty}(\mathbb{R}^2)$ ...

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**22**answers

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### Which high-degree derivatives play an essential role?

Q. Which high-degree derivatives play an essential role
in applications, or in theorems?
Of course the 1st derivative of distance w.r.t. time (velocity), the 2nd derivative (acceleration),
and the ...

**2**

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**0**answers

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### Inequality for integro-differential system

Reading a paper about homogenization of stochastic coefficients [A. Gloria et al., Invent. math. (2015)], I found the following lemma that gives an estimate of the solution for a given ODE system. The ...

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### Limits of the wave equation with piecewise constant propagation speed

This question is cross-posted from math.stackexchange.com, where it did not (yet?) get any answers despite a +100 bounty.
Consider a wave equation
$$\frac{\partial^2 u}{\partial t^2} = c(x)^2 \frac{\...

**0**

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**0**answers

51 views

### $L^1$ estimate of the heat equation

The classical heat equation read $\partial_tu-\Delta u=0$ in $\Bbb R^3$. If $u$ is sufficintly smooth, then we can multiply both sides by $|u|^{p-2}u$ to get the $L^p$ estimate, where $1<p<\...

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275 views

### On Solving a Fourth-Order Non-Linear PDE

I am presently working on a problem in fluid dynamics where our group is investigating the behavior of temperature and velocity at the leading edge of a flat plate when fluid flows past it. The ...

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44 views

### Lax Pair for Kadomtsev-Petviashvili Equation (2-dim KdV)

Is there a simple way to show that the pair
\begin{equation}\begin{cases}\alpha \psi_y+\psi_{xx}+u\psi=0 \\\psi_t + 4\psi_{xxx}+6u\psi_x+3u_x\psi -3\alpha (\partial_x^{-1}u_y)\psi=0\end{cases}\end{...

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### Underspecified Riccatti-type ODE

I came across the following Ricatti-type ODE in my reading
$$
\begin{aligned}
\partial_t \psi(t,x) &= \Psi(\psi(t,x)),\\
\psi(0,x)&=x,\\
\Psi(x)&\triangleq \partial_t\psi(t,x)|_{t=0^+}.
\...

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49 views

### Riccati equations for dealing with stiffness of ODEs

Early texts [Forman Acton, Numerical Methods that Work; Numerical Recipes, first edition] suggested that for integrating a stiff linear ODE it was very helpful to integrate the associated Riccati ...